Solve 7/x^2-4-5/x^2+4x+4

$$\frac { 7 } { x ^ { 2 } - 4 } - \frac { 5 } { x ^ { 2 } + 4 x + 4 }$$

Evaluate

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

Short Solution Steps

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

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

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{7}{\left(x-2\right)\left(x+2\right)} times \frac{x+2}{x+2}. Multiply \frac{5}{\left(x+2\right)^{2}} times \frac{x-2}{x-2}.

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

Since \frac{7\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}} and \frac{5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 7\left(x+2\right)-5\left(x-2\right).

$$\frac{7x+14-5x+10}{\left(x-2\right)\left(x+2\right)^{2}}$$

Combine like terms in 7x+14-5x+10.

$$\frac{2x+24}{\left(x-2\right)\left(x+2\right)^{2}}$$

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

$$\frac{2x+24}{x^{3}+2x^{2}-4x-8}$$

Differentiate w.r.t. x

$$-\frac{\left(2x+17\right)^{2}-329}{\left(x-2\right)^{2}\left(x+2\right)^{3}}$$