Question

what is the difference?
\frac{x}{x^{2}-2x - 15}-\frac{4}{x^{2}+2x - 35}
\frac{x^{2}+3x + 12}{(x - 3)(x - 5)(x + 7)}
\frac{x(x + 3-12)}{(x + 3)(x - 5)(x + 7)}
\frac{x^{2}+3x + 12}{(x + 3)(x - 5)(x + 7)}
\frac{x^{2}+3x - 12}{(x + 3)(x - 5)(x + 7)}

Explanation:

Step1: Factor the denominators

Factor $x^{2}-2x - 15=(x - 5)(x+3)$ and $x^{2}+2x - 35=(x + 7)(x - 5)$.

Step2: Find the common - denominator

The common denominator of $\frac{x}{x^{2}-2x - 15}$ and $\frac{4}{x^{2}+2x - 35}$ is $(x + 3)(x - 5)(x + 7)$.

Step3: Rewrite the fractions with the common denominator

$\frac{x}{x^{2}-2x - 15}=\frac{x(x + 7)}{(x + 3)(x - 5)(x + 7)}=\frac{x^{2}+7x}{(x + 3)(x - 5)(x + 7)}$ and $\frac{4}{x^{2}+2x - 35}=\frac{4(x + 3)}{(x + 3)(x - 5)(x + 7)}=\frac{4x+12}{(x + 3)(x - 5)(x + 7)}$.

Step4: Subtract the fractions

$\frac{x}{x^{2}-2x - 15}-\frac{4}{x^{2}+2x - 35}=\frac{x^{2}+7x-(4x + 12)}{(x + 3)(x - 5)(x + 7)}=\frac{x^{2}+7x-4x - 12}{(x + 3)(x - 5)(x + 7)}=\frac{x^{2}+3x - 12}{(x + 3)(x - 5)(x + 7)}$

Answer:

$\frac{x^{2}+3x - 12}{(x + 3)(x - 5)(x + 7)}$ (the last option)