Solve 1/x^2-7-2/(x+1)^2

$$\frac { 1 } { x ^ { 2 } - 7 } - \frac { 2 } { ( x + 1 ) ^ { 2 } }$$

Evaluate

$$-\frac{\left(x-5\right)\left(x+3\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}-7 and \left(x+1\right)^{2} is \left(x^{2}-7\right)\left(x+1\right)^{2}. Multiply \frac{1}{x^{2}-7} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{2}{\left(x+1\right)^{2}} times \frac{x^{2}-7}{x^{2}-7}.

$$\frac{\left(x+1\right)^{2}}{\left(x^{2}-7\right)\left(x+1\right)^{2}}-\frac{2\left(x^{2}-7\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Since \frac{\left(x+1\right)^{2}}{\left(x^{2}-7\right)\left(x+1\right)^{2}} and \frac{2\left(x^{2}-7\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)^{2}-2\left(x^{2}-7\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Do the multiplications in \left(x+1\right)^{2}-2\left(x^{2}-7\right).

$$\frac{x^{2}+2x+1-2x^{2}+14}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Combine like terms in x^{2}+2x+1-2x^{2}+14.

$$\frac{-x^{2}+2x+15}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Expand \left(x^{2}-7\right)\left(x+1\right)^{2}.

$$\frac{-x^{2}+2x+15}{x^{4}+2x^{3}-6x^{2}-14x-7}$$

Expand

$$\frac{15+2x-x^{2}}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}-7 and \left(x+1\right)^{2} is \left(x^{2}-7\right)\left(x+1\right)^{2}. Multiply \frac{1}{x^{2}-7} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{2}{\left(x+1\right)^{2}} times \frac{x^{2}-7}{x^{2}-7}.

$$\frac{\left(x+1\right)^{2}}{\left(x^{2}-7\right)\left(x+1\right)^{2}}-\frac{2\left(x^{2}-7\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Since \frac{\left(x+1\right)^{2}}{\left(x^{2}-7\right)\left(x+1\right)^{2}} and \frac{2\left(x^{2}-7\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)^{2}-2\left(x^{2}-7\right)}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Do the multiplications in \left(x+1\right)^{2}-2\left(x^{2}-7\right).

$$\frac{x^{2}+2x+1-2x^{2}+14}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Combine like terms in x^{2}+2x+1-2x^{2}+14.

$$\frac{-x^{2}+2x+15}{\left(x^{2}-7\right)\left(x+1\right)^{2}}$$

Expand \left(x^{2}-7\right)\left(x+1\right)^{2}.

$$\frac{-x^{2}+2x+15}{x^{4}+2x^{3}-6x^{2}-14x-7}$$