Solve 4/x^2+7x+3/x^2+5x-14

$$\frac { 4 } { x ^ { 2 } + 7 x } + \frac { 3 } { x ^ { 2 } + 5 x - 14 }$$

Evaluate

$$\frac{7x-8}{x\left(x-2\right)\left(x+7\right)}$$

Short Solution Steps

Factor x^{2}+7x. Factor x^{2}+5x-14.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+7\right) and \left(x-2\right)\left(x+7\right) is x\left(x-2\right)\left(x+7\right). Multiply \frac{4}{x\left(x+7\right)} times \frac{x-2}{x-2}. Multiply \frac{3}{\left(x-2\right)\left(x+7\right)} times \frac{x}{x}.

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

Since \frac{4\left(x-2\right)}{x\left(x-2\right)\left(x+7\right)} and \frac{3x}{x\left(x-2\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

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

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

$$\frac{4x-8+3x}{x\left(x-2\right)\left(x+7\right)}$$

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

$$\frac{7x-8}{x\left(x-2\right)\left(x+7\right)}$$

Expand x\left(x-2\right)\left(x+7\right).

$$\frac{7x-8}{x^{3}+5x^{2}-14x}$$

Differentiate w.r.t. x

$$\frac{-14x^{3}-11x^{2}+80x-112}{\left(x\left(x-2\right)\left(x+7\right)\right)^{2}}$$