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

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

Evaluate

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

Short Solution Steps

Calculate 2 to the power of 2 and get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x-2 is \left(x-2\right)\left(x+2\right). Multiply \frac{3}{x+2} times \frac{x-2}{x-2}. Multiply \frac{4}{x-2} times \frac{x+2}{x+2}.

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

Since \frac{3\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{4\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x-2\right)-4\left(x+2\right).

$$\frac{3x-6-4x-8}{\left(x-2\right)\left(x+2\right)}+\frac{4}{x^{2}+4x+4}$$

Combine like terms in 3x-6-4x-8.

$$\frac{-x-14}{\left(x-2\right)\left(x+2\right)}+\frac{4}{x^{2}+4x+4}$$

Factor x^{2}+4x+4.

$$\frac{-x-14}{\left(x-2\right)\left(x+2\right)}+\frac{4}{\left(x+2\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and \left(x+2\right)^{2} is \left(x-2\right)\left(x+2\right)^{2}. Multiply \frac{-x-14}{\left(x-2\right)\left(x+2\right)} times \frac{x+2}{x+2}. Multiply \frac{4}{\left(x+2\right)^{2}} times \frac{x-2}{x-2}.

$$\frac{\left(-x-14\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}}+\frac{4\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}}$$

Since \frac{\left(-x-14\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}} and \frac{4\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-x-14\right)\left(x+2\right)+4\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}}$$

Do the multiplications in \left(-x-14\right)\left(x+2\right)+4\left(x-2\right).

$$\frac{-x^{2}-2x-14x-28+4x-8}{\left(x-2\right)\left(x+2\right)^{2}}$$

Combine like terms in -x^{2}-2x-14x-28+4x-8.

$$\frac{-x^{2}-12x-36}{\left(x-2\right)\left(x+2\right)^{2}}$$

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

$$\frac{-x^{2}-12x-36}{x^{3}+2x^{2}-4x-8}$$

Differentiate w.r.t. x

$$\frac{\left(x+6\right)\left(x^{2}+16x-4\right)}{\left(x-2\right)^{2}\left(x+2\right)^{3}}$$