Solve x+1+3x/x^2-x-6

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

Evaluate

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

Short Solution Steps

Factor x^{2}-x-6.

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

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

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

Since \frac{\left(x+1\right)\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+2\right)} and \frac{3x}{\left(x-3\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x+1\right)\left(x-3\right)\left(x+2\right)+3x.

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

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

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

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

$$\frac{x^{3}-4x-6}{x^{2}-x-6}$$

Differentiate w.r.t. x

$$\frac{x^{4}-2x^{3}-14x^{2}+12x+18}{\left(\left(x-3\right)\left(x+2\right)\right)^{2}}$$