Solve 4x/x^2-4+2/x^2+x-6

$$\frac { 4 x } { x ^ { 2 } - 4 } + \frac { 2 } { x ^ { 2 } + x - 6 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-4. Factor x^{2}+x-6.

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

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)\left(x+3\right) is \left(x-2\right)\left(x+2\right)\left(x+3\right). Multiply \frac{4x}{\left(x-2\right)\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{2}{\left(x-2\right)\left(x+3\right)} times \frac{x+2}{x+2}.

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

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

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

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

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

Combine like terms in 4x^{2}+12x+2x+4.

$$\frac{4x^{2}+14x+4}{\left(x-2\right)\left(x+2\right)\left(x+3\right)}$$

Expand \left(x-2\right)\left(x+2\right)\left(x+3\right).

$$\frac{4x^{2}+14x+4}{x^{3}+3x^{2}-4x-12}$$

Factor

$$\frac{4\left(x-\frac{-\sqrt{33}-7}{4}\right)\left(x-\frac{\sqrt{33}-7}{4}\right)}{\left(x+3\right)\left(x^{2}-4\right)}$$