Solve 4/x^2+2x-8-5/x^2+3x-10

$$\frac { 4 } { x ^ { 2 } + 2 x - 8 } - \frac { 5 } { x ^ { 2 } + 3 x - 10 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+2x-8. Factor x^{2}+3x-10.

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

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

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

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

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

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

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

Combine like terms in 4x+20-5x-20.

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

Expand \left(x-2\right)\left(x+4\right)\left(x+5\right).

$$\frac{-x}{x^{3}+7x^{2}+2x-40}$$

Differentiate w.r.t. x

$$\frac{2x^{3}+7x^{2}+40}{\left(\left(x-2\right)\left(x+4\right)\left(x+5\right)\right)^{2}}$$