Solve 1/x^2+4x+4-1/x^2-3x-10

$$\frac { 1 } { x ^ { 2 } + 4 x + 4 } - \frac { 1 } { x ^ { 2 } - 3 x - 10 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+4x+4. Factor x^{2}-3x-10.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)^{2} and \left(x-5\right)\left(x+2\right) is \left(x-5\right)\left(x+2\right)^{2}. Multiply \frac{1}{\left(x+2\right)^{2}} times \frac{x-5}{x-5}. Multiply \frac{1}{\left(x-5\right)\left(x+2\right)} times \frac{x+2}{x+2}.

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

Since \frac{x-5}{\left(x-5\right)\left(x+2\right)^{2}} and \frac{x+2}{\left(x-5\right)\left(x+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x-5-\left(x+2\right).

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

Combine like terms in x-5-x-2.

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

Expand \left(x-5\right)\left(x+2\right)^{2}.

$$\frac{-7}{x^{3}-x^{2}-16x-20}$$

Differentiate w.r.t. x

$$\frac{7\left(3x-8\right)}{\left(x-5\right)^{2}\left(x+2\right)^{3}}$$