Solve 4/x^2-3x-10+x/x^2-9x+20

$$\frac { 4 } { x ^ { 2 } - 3 x - 10 } + \frac { x } { x ^ { 2 } - 9 x + 20 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-3x-10. Factor x^{2}-9x+20.

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

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

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

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

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

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

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

Combine like terms in 4x-16+x^{2}+2x.

$$\frac{6x-16+x^{2}}{\left(x-5\right)\left(x-4\right)\left(x+2\right)}$$

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

$$\frac{6x-16+x^{2}}{x^{3}-7x^{2}+2x+40}$$

Differentiate w.r.t. x

$$\frac{272-144x+92x^{2}-12x^{3}-x^{4}}{\left(\left(x-5\right)\left(x-4\right)\left(x+2\right)\right)^{2}}$$