Simplify 3/(x^2-4)-(x+5)/(x+2)

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

Step 1

Simplify the denominator.

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

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

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 $b = 2$ .

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

Step 2

To write $- \frac{x + 5}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

Simplify terms.

Multiply $\frac{x + 5}{x + 2}$ by $\frac{x - 2}{x - 2}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{3 + (- x - 1 \cdot 5) (x - 2)}{(x + 2) (x - 2)}$

Multiply $- 1$ by $5$ .

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

Expand $(- x - 5) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- 5$ by $- 2$ .

$\frac{3 - x^{2} + 2 x - 5 x + 10}{(x + 2) (x - 2)}$

Subtract $5 x$ from $2 x$ .

$\frac{3 - x^{2} - 3 x + 10}{(x + 2) (x - 2)}$

Add $3$ and $10$ .

$\frac{- x^{2} - 3 x + 13}{(x + 2) (x - 2)}$

Step 5

Simplify with factoring out.

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) - 3 x + 13}{(x + 2) (x - 2)}$

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

$\frac{- (x^{2}) - (3 x) + 13}{(x + 2) (x - 2)}$

Factor $- 1$ out of $- (x^{2}) - (3 x)$ .

$\frac{- (x^{2} + 3 x) + 13}{(x + 2) (x - 2)}$

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (x^{2} + 3 x) - 1 (- 13)}{(x + 2) (x - 2)}$

Factor $- 1$ out of $- (x^{2} + 3 x) - 1 (- 13)$ .

$\frac{- (x^{2} + 3 x - 13)}{(x + 2) (x - 2)}$

Simplify the expression.

Rewrite $- (x^{2} + 3 x - 13)$ as $- 1 (x^{2} + 3 x - 13)$ .

$\frac{- 1 (x^{2} + 3 x - 13)}{(x + 2) (x - 2)}$

Move the negative in front of the fraction.

$- \frac{x^{2} + 3 x - 13}{(x + 2) (x - 2)}$

Simplify 3/(x^2-4)-(x+5)/(x+2)

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