Solve -5/x+4+5/x^2+8x+16-1/x^2-16

$$\frac { - 5 } { x + 4 } + \frac { 5 } { x ^ { 2 } + 8 x + 16 } - \frac { 1 } { x ^ { 2 } - 16 }$$

Evaluate

$$\frac{56+4x-5x^{2}}{\left(x-4\right)\left(x+4\right)^{2}}$$

Short Solution Steps

Factor x^{2}+8x+16.

$$\frac{-5}{x+4}+\frac{5}{\left(x+4\right)^{2}}-\frac{1}{x^{2}-16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+4 and \left(x+4\right)^{2} is \left(x+4\right)^{2}. Multiply \frac{-5}{x+4} times \frac{x+4}{x+4}.

$$\frac{-5\left(x+4\right)}{\left(x+4\right)^{2}}+\frac{5}{\left(x+4\right)^{2}}-\frac{1}{x^{2}-16}$$

Since \frac{-5\left(x+4\right)}{\left(x+4\right)^{2}} and \frac{5}{\left(x+4\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-5\left(x+4\right)+5}{\left(x+4\right)^{2}}-\frac{1}{x^{2}-16}$$

Do the multiplications in -5\left(x+4\right)+5.

$$\frac{-5x-20+5}{\left(x+4\right)^{2}}-\frac{1}{x^{2}-16}$$

Combine like terms in -5x-20+5.

$$\frac{-5x-15}{\left(x+4\right)^{2}}-\frac{1}{x^{2}-16}$$

Factor x^{2}-16.

$$\frac{-5x-15}{\left(x+4\right)^{2}}-\frac{1}{\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+4\right)^{2} and \left(x-4\right)\left(x+4\right) is \left(x-4\right)\left(x+4\right)^{2}. Multiply \frac{-5x-15}{\left(x+4\right)^{2}} times \frac{x-4}{x-4}. Multiply \frac{1}{\left(x-4\right)\left(x+4\right)} times \frac{x+4}{x+4}.

$$\frac{\left(-5x-15\right)\left(x-4\right)}{\left(x-4\right)\left(x+4\right)^{2}}-\frac{x+4}{\left(x-4\right)\left(x+4\right)^{2}}$$

Since \frac{\left(-5x-15\right)\left(x-4\right)}{\left(x-4\right)\left(x+4\right)^{2}} and \frac{x+4}{\left(x-4\right)\left(x+4\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-5x-15\right)\left(x-4\right)-\left(x+4\right)}{\left(x-4\right)\left(x+4\right)^{2}}$$

Do the multiplications in \left(-5x-15\right)\left(x-4\right)-\left(x+4\right).

$$\frac{-5x^{2}+20x-15x+60-x-4}{\left(x-4\right)\left(x+4\right)^{2}}$$

Combine like terms in -5x^{2}+20x-15x+60-x-4.

$$\frac{-5x^{2}+4x+56}{\left(x-4\right)\left(x+4\right)^{2}}$$

Expand \left(x-4\right)\left(x+4\right)^{2}.

$$\frac{-5x^{2}+4x+56}{x^{3}+4x^{2}-16x-64}$$

Differentiate w.r.t. x

$$\frac{5x^{3}-28x^{2}+8x+160}{\left(x-4\right)^{2}\left(x+4\right)^{3}}$$