Solve x+1/x^2+x-12-x-4/x^2+7x+12

$$\frac { x + 1 } { x ^ { 2 } + x - 12 } - \frac { x - 4 } { x ^ { 2 } + 7 x + 12 }$$

Evaluate

$$\frac{11x-9}{\left(x+4\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}+x-12. Factor x^{2}+7x+12.

$$\frac{x+1}{\left(x-3\right)\left(x+4\right)}-\frac{x-4}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and \left(x+3\right)\left(x+4\right) is \left(x-3\right)\left(x+3\right)\left(x+4\right). Multiply \frac{x+1}{\left(x-3\right)\left(x+4\right)} times \frac{x+3}{x+3}. Multiply \frac{x-4}{\left(x+3\right)\left(x+4\right)} times \frac{x-3}{x-3}.

$$\frac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}-\frac{\left(x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)} and \frac{\left(x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\left(x+3\right)-\left(x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in \left(x+1\right)\left(x+3\right)-\left(x-4\right)\left(x-3\right).

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

Combine like terms in x^{2}+3x+x+3-x^{2}+3x+4x-12.

$$\frac{11x-9}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Expand \left(x-3\right)\left(x+3\right)\left(x+4\right).

$$\frac{11x-9}{x^{3}+4x^{2}-9x-36}$$

Expand

$$\frac{11x-9}{\left(x+4\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}+x-12. Factor x^{2}+7x+12.

$$\frac{x+1}{\left(x-3\right)\left(x+4\right)}-\frac{x-4}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and \left(x+3\right)\left(x+4\right) is \left(x-3\right)\left(x+3\right)\left(x+4\right). Multiply \frac{x+1}{\left(x-3\right)\left(x+4\right)} times \frac{x+3}{x+3}. Multiply \frac{x-4}{\left(x+3\right)\left(x+4\right)} times \frac{x-3}{x-3}.

$$\frac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}-\frac{\left(x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)} and \frac{\left(x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\left(x+3\right)-\left(x-4\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in \left(x+1\right)\left(x+3\right)-\left(x-4\right)\left(x-3\right).

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

Combine like terms in x^{2}+3x+x+3-x^{2}+3x+4x-12.

$$\frac{11x-9}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Expand \left(x-3\right)\left(x+3\right)\left(x+4\right).

$$\frac{11x-9}{x^{3}+4x^{2}-9x-36}$$