Solve 1/x^2+x-6-2/x-2+4/x-4

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

Evaluate

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

Short Solution Steps

Factor x^{2}+x-6.

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

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

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

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

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

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

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

Combine like terms in 1-2x-6.

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

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

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

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

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

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

$$\frac{-5x+20-2x^{2}+8x+4x^{2}+12x-8x-24}{\left(x-4\right)\left(x-2\right)\left(x+3\right)}$$

Combine like terms in -5x+20-2x^{2}+8x+4x^{2}+12x-8x-24.

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

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

$$\frac{7x-4+2x^{2}}{x^{3}-3x^{2}-10x+24}$$

Differentiate w.r.t. x

$$\frac{128+72x+13x^{2}-14x^{3}-2x^{4}}{\left(\left(x-4\right)\left(x-2\right)\left(x+3\right)\right)^{2}}$$