Simplify (4x)/(x^2-9)-6/(x^2+x-12)

$\frac{4 x}{x^{2} - 9} - \frac{6}{x^{2} + x - 12}$

Simplify each term.

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Factor $2$ out of $4 x (x + 4) - 6 (x + 3)$ .

Factor $2$ out of $4 x (x + 4)$ .

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

Factor $2$ out of $- 6 (x + 3)$ .

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

Factor $2$ out of $2 (2 x (x + 4)) + 2 (- 3 (x + 3))$ .

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

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $4$ by $2$ .

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

Apply the distributive property.

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

Multiply $- 3$ by $3$ .

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

Subtract $3 x$ from $8 x$ .

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

Simplify (4x)/(x^2-9)-6/(x^2+x-12)

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