Simplify (2x)/(4x^2+9x+2)+3/(2x^2-8x-24)

$\frac{2 x}{4 x^{2} + 9 x + 2} + \frac{3}{2 x^{2} - 8 x - 24}$

Step 1

Simplify each term.

Factor by grouping.

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 $a \cdot c = 4 \cdot 2 = 8$ and whose sum is $b = 9$ .

Factor $9$ out of $9 x$ .

$\frac{2 x}{4 x^{2} + 9 (x) + 2} + \frac{3}{2 x^{2} - 8 x - 24}$

Rewrite $9$ as $1$ plus $8$

$\frac{2 x}{4 x^{2} + (1 + 8) x + 2} + \frac{3}{2 x^{2} - 8 x - 24}$

Apply the distributive property.

$\frac{2 x}{4 x^{2} + 1 x + 8 x + 2} + \frac{3}{2 x^{2} - 8 x - 24}$

Multiply $x$ by $1$ .

$\frac{2 x}{4 x^{2} + x + 8 x + 2} + \frac{3}{2 x^{2} - 8 x - 24}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{2 x}{(4 x^{2} + x) + 8 x + 2} + \frac{3}{2 x^{2} - 8 x - 24}$

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

$\frac{2 x}{x (4 x + 1) + 2 (4 x + 1)} + \frac{3}{2 x^{2} - 8 x - 24}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{2 x}{(4 x + 1) (x + 2)} + \frac{3}{2 x^{2} - 8 x - 24}$

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 8 x - 24$ .

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

$\frac{2 x}{(4 x + 1) (x + 2)} + \frac{3}{2 (x^{2}) - 8 x - 24}$

Factor $2$ out of $- 8 x$ .

$\frac{2 x}{(4 x + 1) (x + 2)} + \frac{3}{2 (x^{2}) + 2 (- 4 x) - 24}$

Factor $2$ out of $- 24$ .

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

Factor $2$ out of $2 x^{2} + 2 (- 4 x)$ .

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

Factor $2$ out of $2 (x^{2} - 4 x) + 2 \cdot - 12$ .

$\frac{2 x}{(4 x + 1) (x + 2)} + \frac{3}{2 (x^{2} - 4 x - 12)}$

Factor $x^{2} - 4 x - 12$ 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 $- 4$ .

$- 6, 2$

Write the factored form using these integers.

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

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

Step 2

To write $\frac{2 x}{(4 x + 1) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{2 (x - 6)}{2 (x - 6)}$ .

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

Step 3

To write $\frac{3}{2 (x - 6) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{4 x + 1}{4 x + 1}$ .

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

Step 4

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

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

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

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

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

Reorder the factors of $(4 x + 1) (x + 2) (2 (x - 6))$ .

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

Reorder the factors of $2 (x - 6) (x + 2) (4 x + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $- 6$ .

$\frac{2 x (2 x - 12) + 3 (4 x + 1)}{2 (4 x + 1) (x + 2) (x - 6)}$

Apply the distributive property.

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

Rewrite using the commutative property of multiplication.

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

Multiply $- 12$ by $2$ .

$\frac{2 \cdot 2 x \cdot x - 24 x + 3 (4 x + 1)}{2 (4 x + 1) (x + 2) (x - 6)}$

Simplify each term.

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

Move $x$ .

$\frac{2 \cdot 2 (x \cdot x) - 24 x + 3 (4 x + 1)}{2 (4 x + 1) (x + 2) (x - 6)}$

Multiply $x$ by $x$ .

$\frac{2 \cdot 2 x^{2} - 24 x + 3 (4 x + 1)}{2 (4 x + 1) (x + 2) (x - 6)}$

Multiply $2$ by $2$ .

$\frac{4 x^{2} - 24 x + 3 (4 x + 1)}{2 (4 x + 1) (x + 2) (x - 6)}$

Apply the distributive property.

$\frac{4 x^{2} - 24 x + 3 (4 x) + 3 \cdot 1}{2 (4 x + 1) (x + 2) (x - 6)}$

Multiply $4$ by $3$ .

$\frac{4 x^{2} - 24 x + 12 x + 3 \cdot 1}{2 (4 x + 1) (x + 2) (x - 6)}$

Multiply $3$ by $1$ .

$\frac{4 x^{2} - 24 x + 12 x + 3}{2 (4 x + 1) (x + 2) (x - 6)}$

Add $- 24 x$ and $12 x$ .

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

Simplify (2x)/(4x^2+9x+2)+3/(2x^2-8x-24)

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