Combine x/(x-3)-3/(x+4)+7/(x^2+x-12)

$\frac{x}{x - 3} - \frac{3}{x + 4} + \frac{7}{x^{2} + x - 12}$

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

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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.

$\frac{x}{x - 3} - \frac{3}{x + 4} + \frac{7}{(x - 3) (x + 4)}$

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

$\frac{x}{x - 3} \cdot \frac{x + 4}{x + 4} - \frac{3}{x + 4} + \frac{7}{(x - 3) (x + 4)}$

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

$\frac{x}{x - 3} \cdot \frac{x + 4}{x + 4} - \frac{3}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{7}{(x - 3) (x + 4)}$

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

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Multiply $\frac{x}{x - 3}$ and $\frac{x + 4}{x + 4}$ .

$\frac{x (x + 4)}{(x - 3) (x + 4)} - \frac{3}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{7}{(x - 3) (x + 4)}$

Multiply $\frac{3}{x + 4}$ and $\frac{x - 3}{x - 3}$ .

$\frac{x (x + 4)}{(x - 3) (x + 4)} - \frac{3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

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

$\frac{x (x + 4)}{(x + 4) (x - 3)} - \frac{3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Combine the numerators over the common denominator.

$\frac{x (x + 4) - 3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Simplify the numerator.

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Apply the distributive property.

$\frac{x \cdot x + x \cdot 4 - 3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 4 - 3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

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

$\frac{x^{2} + 4 \cdot x - 3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Apply the distributive property.

$\frac{x^{2} + 4 x - 3 x - 3 \cdot - 3}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Multiply $- 3$ by $- 3$ .

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

Subtract $3 x$ from $4 x$ .

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

Combine into one fraction.

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

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

Combine the numerators over the common denominator.

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

Add $9$ and $7$ .

$\frac{x^{2} + x + 16}{(x + 4) (x - 3)}$

Combine x/(x-3)-3/(x+4)+7/(x^2+x-12)

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