Solve 3/x^2-x-6+(-1/x(x+4))

$$\frac { 3 } { x ^ { 2 } - x - 6 } + ( - \frac { 1 } { x ( x + 4 ) } )$$

Evaluate

$$\frac{\left(x+6\right)\left(2x+1\right)}{x\left(x-3\right)\left(x+2\right)\left(x+4\right)}$$

Short Solution Steps

Factor x^{2}-x-6.

$$\frac{3}{\left(x-3\right)\left(x+2\right)}-\frac{1}{x\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+2\right) and x\left(x+4\right) is x\left(x-3\right)\left(x+2\right)\left(x+4\right). Multiply \frac{3}{\left(x-3\right)\left(x+2\right)} times \frac{x\left(x+4\right)}{x\left(x+4\right)}. Multiply \frac{1}{x\left(x+4\right)} times \frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+2\right)}.

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

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

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

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

$$\frac{3x^{2}+12x-x^{2}-2x+3x+6}{x\left(x-3\right)\left(x+2\right)\left(x+4\right)}$$

Combine like terms in 3x^{2}+12x-x^{2}-2x+3x+6.

$$\frac{2x^{2}+13x+6}{x\left(x-3\right)\left(x+2\right)\left(x+4\right)}$$

Expand x\left(x-3\right)\left(x+2\right)\left(x+4\right).

$$\frac{2x^{2}+13x+6}{x^{4}+3x^{3}-10x^{2}-24x}$$

Expand

$$\frac{2x^{2}+13x+6}{x\left(x+4\right)\left(x^{2}-x-6\right)}$$

Short Solution Steps

Factor x^{2}-x-6.

$$\frac{3}{\left(x-3\right)\left(x+2\right)}-\frac{1}{x\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+2\right) and x\left(x+4\right) is x\left(x-3\right)\left(x+2\right)\left(x+4\right). Multiply \frac{3}{\left(x-3\right)\left(x+2\right)} times \frac{x\left(x+4\right)}{x\left(x+4\right)}. Multiply \frac{1}{x\left(x+4\right)} times \frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+2\right)}.

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

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

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

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

$$\frac{3x^{2}+12x-x^{2}-2x+3x+6}{x\left(x-3\right)\left(x+2\right)\left(x+4\right)}$$

Combine like terms in 3x^{2}+12x-x^{2}-2x+3x+6.

$$\frac{2x^{2}+13x+6}{x\left(x-3\right)\left(x+2\right)\left(x+4\right)}$$

Expand x\left(x-3\right)\left(x+2\right)\left(x+4\right).

$$\frac{2x^{2}+13x+6}{x^{4}+3x^{3}-10x^{2}-24x}$$