Solve x^2-x-6/x^2-9+x^2+2x-24/x^2-x-12

$$\frac { x ^ { 2 } - x - 6 } { x ^ { 2 } - 9 } + \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - x - 12 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-x-6}{x^{2}-9}.

$$\frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+3\right)}+\frac{x^{2}+2x-24}{x^{2}-x-12}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{x+2}{x+3}+\frac{x^{2}+2x-24}{x^{2}-x-12}$$

Factor the expressions that are not already factored in \frac{x^{2}+2x-24}{x^{2}-x-12}.

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

Cancel out x-4 in both numerator and denominator.

$$\frac{x+2}{x+3}+\frac{x+6}{x+3}$$

Since \frac{x+2}{x+3} and \frac{x+6}{x+3} have the same denominator, add them by adding their numerators.

$$\frac{x+2+x+6}{x+3}$$

Combine like terms in x+2+x+6.

$$\frac{2x+8}{x+3}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-x-6}{x^{2}-9}.

$$\frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+3\right)}+\frac{x^{2}+2x-24}{x^{2}-x-12}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{x+2}{x+3}+\frac{x^{2}+2x-24}{x^{2}-x-12}$$

Factor the expressions that are not already factored in \frac{x^{2}+2x-24}{x^{2}-x-12}.

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

Cancel out x-4 in both numerator and denominator.

$$\frac{x+2}{x+3}+\frac{x+6}{x+3}$$

Since \frac{x+2}{x+3} and \frac{x+6}{x+3} have the same denominator, add them by adding their numerators.

$$\frac{x+2+x+6}{x+3}$$

Combine like terms in x+2+x+6.

$$\frac{2x+8}{x+3}$$