Simplify (x-4)/(x^2+5x+6)+(x-1)/(x^2-4)

$\frac{x - 4}{x^{2} + 5 x + 6} + \frac{x - 1}{x^{2} - 4}$

Simplify each term.

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

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

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $(x + 2) (x - 2) (x + 3)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(x - 4) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $- 4$ by $- 2$ .

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

Subtract $4 x$ from $- 2 x$ .

$\frac{x^{2} - 6 x + 8 + (x - 1) (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Expand $(x - 1) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{x^{2} - 6 x + 8 + x (x + 3) - 1 (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Apply the distributive property.

$\frac{x^{2} - 6 x + 8 + x \cdot x + x \cdot 3 - 1 (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Apply the distributive property.

$\frac{x^{2} - 6 x + 8 + x \cdot x + x \cdot 3 - 1 x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + x \cdot 3 - 1 x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

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

$\frac{x^{2} - 6 x + 8 + x^{2} + 3 \cdot x - 1 x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + 3 x - x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Multiply $- 1$ by $3$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + 3 x - x - 3}{(x + 2) (x + 3) (x - 2)}$

Subtract $x$ from $3 x$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + 2 x - 3}{(x + 2) (x + 3) (x - 2)}$

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

$\frac{2 x^{2} - 6 x + 8 + 2 x - 3}{(x + 2) (x + 3) (x - 2)}$

Add $- 6 x$ and $2 x$ .

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

Subtract $3$ from $8$ .

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

Simplify (x-4)/(x^2+5x+6)+(x-1)/(x^2-4)

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