Simplify (4x-5)/(x^2+x-12)+9/(18-3x-x^2)+2/(x^2+10x+24)

$\frac{4 x - 5}{x^{2} + x - 12} + \frac{9}{18 - 3 x - x^{2}} + \frac{2}{x^{2} + 10 x + 24}$

Simplify each term.

Factor $x^{2} + x - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{18 - 3 x - x^{2}} + \frac{2}{x^{2} + 10 x + 24}$

Factor by grouping.

Reorder terms.

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{- x^{2} - 3 x + 18} + \frac{2}{x^{2} + 10 x + 24}$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 18 = - 18$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{- x^{2} - 3 (x) + 18} + \frac{2}{x^{2} + 10 x + 24}$

Rewrite $- 3$ as $3$ plus $- 6$

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{- x^{2} + (3 - 6) x + 18} + \frac{2}{x^{2} + 10 x + 24}$

Apply the distributive property.

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{- x^{2} + 3 x - 6 x + 18} + \frac{2}{x^{2} + 10 x + 24}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{(- x^{2} + 3 x) - 6 x + 18} + \frac{2}{x^{2} + 10 x + 24}$

Factor out the greatest common factor (GCF) from each group.

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{x (- x + 3) + 6 (- x + 3)} + \frac{2}{x^{2} + 10 x + 24}$

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{(- x + 3) (x + 6)} + \frac{2}{x^{2} + 10 x + 24}$

Factor $x^{2} + 10 x + 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $24$ and whose sum is $10$ .

$4, 6$

Write the factored form using these integers.

$\frac{4 x - 5}{(x - 3) (x + 4)} + \frac{9}{(- x + 3) (x + 6)} + \frac{2}{(x + 4) (x + 6)}$

Find the common denominator.

Multiply $\frac{4 x - 5}{(x - 3) (x + 4)}$ by $\frac{(- x + 3) (x + 6)}{(- x + 3) (x + 6)}$ .

$\frac{4 x - 5}{(x - 3) (x + 4)} \cdot \frac{(- x + 3) (x + 6)}{(- x + 3) (x + 6)} + \frac{9}{(- x + 3) (x + 6)} + \frac{2}{(x + 4) (x + 6)}$

Multiply $\frac{4 x - 5}{(x - 3) (x + 4)}$ by $\frac{(- x + 3) (x + 6)}{(- x + 3) (x + 6)}$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(x - 3) (x + 4) ((- x + 3) (x + 6))} + \frac{9}{(- x + 3) (x + 6)} + \frac{2}{(x + 4) (x + 6)}$

Multiply $\frac{9}{(- x + 3) (x + 6)}$ by $\frac{(x - 3) (x + 4)}{(x - 3) (x + 4)}$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(x - 3) (x + 4) ((- x + 3) (x + 6))} + \frac{9}{(- x + 3) (x + 6)} \cdot \frac{(x - 3) (x + 4)}{(x - 3) (x + 4)} + \frac{2}{(x + 4) (x + 6)}$

Multiply $\frac{9}{(- x + 3) (x + 6)}$ by $\frac{(x - 3) (x + 4)}{(x - 3) (x + 4)}$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(x - 3) (x + 4) ((- x + 3) (x + 6))} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2}{(x + 4) (x + 6)}$

Multiply $\frac{2}{(x + 4) (x + 6)}$ by $\frac{- 1 {(- x + 3)}^{2}}{- 1 {(- x + 3)}^{2}}$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(x - 3) (x + 4) ((- x + 3) (x + 6))} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2}{(x + 4) (x + 6)} \cdot \frac{- 1 {(- x + 3)}^{2}}{- 1 {(- x + 3)}^{2}}$

Multiply $\frac{2}{(x + 4) (x + 6)}$ by $\frac{- 1 {(- x + 3)}^{2}}{- 1 {(- x + 3)}^{2}}$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(x - 3) (x + 4) ((- x + 3) (x + 6))} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Reorder the factors of $(x - 3) (x + 4) ((- x + 3) (x + 6))$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(- x + 3) (x + 4) (x + 6) (x - 3)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Factor $- 1$ out of $- x$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(- (x) + 3) (x + 4) (x + 6) (x - 3)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{(- (x) - 1 (- 3)) (x + 4) (x + 6) (x - 3)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- (x - 3) (x + 4) (x + 6) (x - 3)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 (x - 3) (x + 4) (x + 6) (x - 3)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Raise $x - 3$ to the power of $1$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 ({(x - 3)}^{1} (x - 3)) (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Raise $x - 3$ to the power of $1$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 ({(x - 3)}^{1} {(x - 3)}^{1}) (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{1 + 1} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Add $1$ and $1$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 6) ((x - 3) (x + 4))} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Reorder the factors of $(- x + 3) (x + 6) ((x - 3) (x + 4))$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- x + 3) (x + 4) (x + 6) (x - 3)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Factor $- 1$ out of $- x$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- (x) + 3) (x + 4) (x + 6) (x - 3)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{(- (x) - 1 (- 3)) (x + 4) (x + 6) (x - 3)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- (x - 3) (x + 4) (x + 6) (x - 3)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- 1 (x - 3) (x + 4) (x + 6) (x - 3)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Raise $x - 3$ to the power of $1$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- 1 ({(x - 3)}^{1} (x - 3)) (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Raise $x - 3$ to the power of $1$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- 1 ({(x - 3)}^{1} {(x - 3)}^{1}) (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{1 + 1} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Add $1$ and $1$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})}$

Reorder the factors of $(x + 4) (x + 6) (- 1 {(- x + 3)}^{2})$ .

$\frac{(4 x - 5) ((- x + 3) (x + 6))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Combine the numerators over the common denominator.

$\frac{(4 x - 5) ((- x + 3) (x + 6)) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Simplify each term.

Expand $(- x + 3) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{(4 x - 5) (- x (x + 6) + 3 (x + 6)) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{(4 x - 5) (- x \cdot x - x \cdot 6 + 3 (x + 6)) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{(4 x - 5) (- x \cdot x - x \cdot 6 + 3 x + 3 \cdot 6) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{(4 x - 5) (- (x \cdot x) - x \cdot 6 + 3 x + 3 \cdot 6) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $x$ by $x$ .

$\frac{(4 x - 5) (- x^{2} - x \cdot 6 + 3 x + 3 \cdot 6) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $6$ by $- 1$ .

$\frac{(4 x - 5) (- x^{2} - 6 x + 3 x + 3 \cdot 6) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $3$ by $6$ .

$\frac{(4 x - 5) (- x^{2} - 6 x + 3 x + 18) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Add $- 6 x$ and $3 x$ .

$\frac{(4 x - 5) (- x^{2} - 3 x + 18) + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Expand $(4 x - 5) (- x^{2} - 3 x + 18)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{4 x (- x^{2}) + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot - 1 x \cdot x^{2} + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$\frac{4 \cdot - 1 (x^{2} x) + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{4 \cdot - 1 (x^{2} x^{1}) + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 \cdot - 1 x^{2 + 1} + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Add $2$ and $1$ .

$\frac{4 \cdot - 1 x^{3} + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $4$ by $- 1$ .

$\frac{- 4 x^{3} + 4 x (- 3 x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Rewrite using the commutative property of multiplication.

$\frac{- 4 x^{3} + 4 \cdot - 3 x \cdot x + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{- 4 x^{3} + 4 \cdot - 3 (x \cdot x) + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $x$ by $x$ .

$\frac{- 4 x^{3} + 4 \cdot - 3 x^{2} + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $4$ by $- 3$ .

$\frac{- 4 x^{3} - 12 x^{2} + 4 x \cdot 18 - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $18$ by $4$ .

$\frac{- 4 x^{3} - 12 x^{2} + 72 x - 5 (- x^{2}) - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $- 1$ by $- 5$ .

$\frac{- 4 x^{3} - 12 x^{2} + 72 x + 5 x^{2} - 5 (- 3 x) - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $- 3$ by $- 5$ .

$\frac{- 4 x^{3} - 12 x^{2} + 72 x + 5 x^{2} + 15 x - 5 \cdot 18 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $- 5$ by $18$ .

$\frac{- 4 x^{3} - 12 x^{2} + 72 x + 5 x^{2} + 15 x - 90 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Add $- 12 x^{2}$ and $5 x^{2}$ .

$\frac{- 4 x^{3} - 7 x^{2} + 72 x + 15 x - 90 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Add $72 x$ and $15 x$ .

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 ((x - 3) (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Expand $(x - 3) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x (x + 4) - 3 (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x \cdot x + x \cdot 4 - 3 (x + 4))}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x \cdot x + x \cdot 4 - 3 x - 3 \cdot 4)}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x^{2} + x \cdot 4 - 3 x - 3 \cdot 4)}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Move $4$ to the left of $x$ .

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x^{2} + 4 \cdot x - 3 x - 3 \cdot 4)}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $- 3$ by $4$ .

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x^{2} + 4 x - 3 x - 12)}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Subtract $3 x$ from $4 x$ .

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 (x^{2} + x - 12)}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 x^{2} + 9 x + 9 \cdot - 12}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Multiply $9$ by $- 12$ .

$\frac{- 4 x^{3} - 7 x^{2} + 87 x - 90 + 9 x^{2} + 9 x - 108}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Simplify by adding terms.

Add $- 7 x^{2}$ and $9 x^{2}$ .

$\frac{- 4 x^{3} + 2 x^{2} + 87 x - 90 + 9 x - 108}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Add $87 x$ and $9 x$ .

$\frac{- 4 x^{3} + 2 x^{2} + 96 x - 90 - 108}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Subtract $108$ from $- 90$ .

$\frac{- 4 x^{3} + 2 x^{2} + 96 x - 198}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Simplify each term.

Factor $2$ out of $- 4 x^{3} + 2 x^{2} + 96 x - 198$ .

Factor $2$ out of $- 4 x^{3}$ .

$\frac{2 (- 2 x^{3}) + 2 x^{2} + 96 x - 198}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (- 2 x^{3}) + 2 (x^{2}) + 96 x - 198}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $96 x$ .

$\frac{2 (- 2 x^{3}) + 2 (x^{2}) + 2 (48 x) - 198}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $- 198$ .

$\frac{2 (- 2 x^{3}) + 2 x^{2} + 2 (48 x) + 2 \cdot - 99}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 (- 2 x^{3}) + 2 x^{2}$ .

$\frac{2 (- 2 x^{3} + x^{2}) + 2 (48 x) + 2 \cdot - 99}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 (- 2 x^{3} + x^{2}) + 2 (48 x)$ .

$\frac{2 (- 2 x^{3} + x^{2} + 48 x) + 2 \cdot - 99}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 (- 2 x^{3} + x^{2} + 48 x) + 2 \cdot - 99$ .

$\frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{- 1 {(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Move the negative in front of the fraction.

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{({(x - 3)}^{2} (x + 4)) (x + 6)} + \frac{2 (- 1 {(- x + 3)}^{2})}{- {(- x + 3)}^{2} (x + 4) (x + 6)}$

Dividing two negative values results in a positive value.

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{{(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 ({(- x + 3)}^{2})}{({(- x + 3)}^{2} (x + 4)) (x + 6)}$

Cancel the common factor of ${(- x + 3)}^{2}$ .

Cancel the common factor.

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{{(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 {(- x + 3)}^{2}}{{(- x + 3)}^{2} (x + 4) (x + 6)}$

Rewrite the expression.

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{{(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2}{(x + 4) (x + 6)}$

To write $\frac{2}{(x + 4) (x + 6)}$ as a fraction with a common denominator, multiply by $\frac{{(x - 3)}^{2}}{{(x - 3)}^{2}}$ .

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{{(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2}{(x + 4) (x + 6)} \cdot \frac{{(x - 3)}^{2}}{{(x - 3)}^{2}}$

Write each expression with a common denominator of $(x + 4) (x + 6) {(x - 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{(x + 4) (x + 6)}$ by $\frac{{(x - 3)}^{2}}{{(x - 3)}^{2}}$ .

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{{(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 {(x - 3)}^{2}}{(x + 4) (x + 6) {(x - 3)}^{2}}$

Reorder the factors of $(x + 4) (x + 6) {(x - 3)}^{2}$ .

$- \frac{2 (- 2 x^{3} + x^{2} + 48 x - 99)}{{(x - 3)}^{2} (x + 4) (x + 6)} + \frac{2 {(x - 3)}^{2}}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Combine the numerators over the common denominator.

$\frac{- 2 (- 2 x^{3} + x^{2} + 48 x - 99) + 2 {(x - 3)}^{2}}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Simplify the numerator.

Factor $2$ out of $- 2 (- 2 x^{3} + x^{2} + 48 x - 99) + 2 {(x - 3)}^{2}$ .

Factor $2$ out of $- 2 (- 2 x^{3} + x^{2} + 48 x - 99)$ .

$\frac{2 (- (- 2 x^{3} + x^{2} + 48 x - 99)) + 2 {(x - 3)}^{2}}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 {(x - 3)}^{2}$ .

$\frac{2 (- (- 2 x^{3} + x^{2} + 48 x - 99)) + 2 ({(x - 3)}^{2})}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 (- (- 2 x^{3} + x^{2} + 48 x - 99)) + 2 ({(x - 3)}^{2})$ .

$\frac{2 (- (- 2 x^{3} + x^{2} + 48 x - 99) + {(x - 3)}^{2})}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{2 (- (- 2 x^{3}) - x^{2} - (48 x) - - 99 + {(x - 3)}^{2})}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Simplify.

Multiply $- 2$ by $- 1$ .

$\frac{2 (2 x^{3} - x^{2} - (48 x) - - 99 + {(x - 3)}^{2})}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Multiply $48$ by $- 1$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x - - 99 + {(x - 3)}^{2})}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Multiply $- 1$ by $- 99$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + {(x - 3)}^{2})}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + (x - 3) (x - 3))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x (x - 3) - 3 (x - 3))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x \cdot x + x \cdot - 3 - 3 (x - 3))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Apply the distributive property.

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Move $- 3$ to the left of $x$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Multiply $- 3$ by $- 3$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x^{2} - 3 x - 3 x + 9)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Subtract $3 x$ from $- 3 x$ .

$\frac{2 (2 x^{3} - x^{2} - 48 x + 99 + x^{2} - 6 x + 9)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Add $- x^{2}$ and $x^{2}$ .

$\frac{2 (2 x^{3} - 48 x + 99 + 0 - 6 x + 9)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Add $2 x^{3}$ and $0$ .

$\frac{2 (2 x^{3} - 48 x + 99 - 6 x + 9)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Subtract $6 x$ from $- 48 x$ .

$\frac{2 (2 x^{3} - 54 x + 99 + 9)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Add $99$ and $9$ .

$\frac{2 (2 x^{3} - 54 x + 108)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Rewrite $2 x^{3} - 54 x + 108$ in a factored form.

Factor $2$ out of $2 x^{3} - 54 x + 108$ .

Factor $2$ out of $2 x^{3}$ .

$\frac{2 (2 (x^{3}) - 54 x + 108)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $- 54 x$ .

$\frac{2 (2 (x^{3}) + 2 (- 27 x) + 108)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $108$ .

$\frac{2 (2 x^{3} + 2 (- 27 x) + 2 \cdot 54)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 x^{3} + 2 (- 27 x)$ .

$\frac{2 (2 (x^{3} - 27 x) + 2 \cdot 54)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $2$ out of $2 (x^{3} - 27 x) + 2 \cdot 54$ .

$\frac{2 (2 (x^{3} - 27 x + 54))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $x^{3} - 27 x + 54$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9 q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9$

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

Substitute $3$ into the polynomial.

$3^{3} - 27 \cdot 3 + 54$

Raise $3$ to the power of $3$ .

$27 - 27 \cdot 3 + 54$

Multiply $- 27$ by $3$ .

$27 - 81 + 54$

Subtract $81$ from $27$ .

$- 54 + 54$

Add $- 54$ and $54$ .

$0$

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 27 x + 54}{x - 3}$

Divide $x^{3} - 27 x + 54$ by $x - 3$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{3}$

$0 x^{2}$

$27 x$

$54$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			+	$x^{3}$	-	$3 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 3 x^{2}$

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					+	$3 x^{2}$	-	$9 x$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 9 x$

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$

Divide the highest order term in the dividend $- 18 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$
							-	$18 x$	+	$54$

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x + 54$

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$
							+	$18 x$	-	$54$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$
							+	$18 x$	-	$54$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 3 x - 18$

Write $x^{3} - 27 x + 54$ as a set of factors.

$\frac{2 (2 ((x - 3) (x^{2} + 3 x - 18)))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Factor $x^{2} + 3 x - 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $3$ .

$- 3, 6$

Write the factored form using these integers.

$\frac{2 (2 ((x - 3) ((x - 3) (x + 6))))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

$\frac{2 (2 ((x - 3) (x - 3) (x + 6)))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Combine like factors.

Raise $x - 3$ to the power of $1$ .

$\frac{2 (2 ({(x - 3)}^{1} (x - 3) (x + 6)))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Raise $x - 3$ to the power of $1$ .

$\frac{2 (2 ({(x - 3)}^{1} {(x - 3)}^{1} (x + 6)))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 (2 ({(x - 3)}^{1 + 1} (x + 6)))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Add $1$ and $1$ .

$\frac{2 (2 ({(x - 3)}^{2} (x + 6)))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

$\frac{2 (2 {(x - 3)}^{2} (x + 6))}{{(x - 3)}^{2} (x + 4) (x + 6)}$

$\frac{2 \cdot 2 {(x - 3)}^{2} (x + 6)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Multiply $2$ by $2$ .

$\frac{4 {(x - 3)}^{2} (x + 6)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of ${(x - 3)}^{2}$ .

Cancel the common factor.

$\frac{4 {(x - 3)}^{2} (x + 6)}{{(x - 3)}^{2} (x + 4) (x + 6)}$

Rewrite the expression.

$\frac{4 (x + 6)}{(x + 4) (x + 6)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{4 (x + 6)}{(x + 4) (x + 6)}$

Rewrite the expression.

$\frac{4}{x + 4}$

Simplify (4x-5)/(x^2+x-12)+9/(18-3x-x^2)+2/(x^2+10x+24)

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