Solve 5x/x^2-9+2/x+4

$$\frac { 5 x } { x ^ { 2 } - 9 } + \frac { 2 } { x + 4 }$$

Evaluate

$$\frac{7x^{2}+20x-18}{\left(x+4\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}-9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right) and x+4 is \left(x-3\right)\left(x+3\right)\left(x+4\right). Multiply \frac{5x}{\left(x-3\right)\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{2}{x+4} times \frac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}.

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

Since \frac{5x\left(x+4\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)} and \frac{2\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 5x\left(x+4\right)+2\left(x-3\right)\left(x+3\right).

$$\frac{5x^{2}+20x+2x^{2}+6x-6x-18}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 5x^{2}+20x+2x^{2}+6x-6x-18.

$$\frac{7x^{2}+20x-18}{\left(x-3\right)\left(x+3\right)\left(x+4\right)}$$

Expand \left(x-3\right)\left(x+3\right)\left(x+4\right).

$$\frac{7x^{2}+20x-18}{x^{3}+4x^{2}-9x-36}$$

Differentiate w.r.t. x

$$-\frac{7x^{4}+40x^{3}+89x^{2}+360x+882}{\left(\left(x+4\right)\left(x^{2}-9\right)\right)^{2}}$$