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

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

Evaluate

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

Short Solution Steps

Factor 6+x-x^{2}.

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

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

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

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

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

Do the multiplications in 3\left(-x-2\right)-x^{2}.

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

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+2 is \left(x-3\right)\left(x+2\right). Multiply \frac{-3x-6-x^{2}}{\left(x-3\right)\left(-x-2\right)} times \frac{-1}{-1}. Multiply \frac{2}{x+2} times \frac{x-3}{x-3}.

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

Since \frac{-\left(-3x-6-x^{2}\right)}{\left(x-3\right)\left(x+2\right)} and \frac{2\left(x-3\right)}{\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(-3x-6-x^{2}\right)-2\left(x-3\right).

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

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

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

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

$$\frac{x+12+x^{2}}{x^{2}-x-6}$$

Differentiate w.r.t. x

$$\frac{2\left(3-18x-x^{2}\right)}{x^{4}-2x^{3}-11x^{2}+12x+36}$$