Solve x^2+x-2/x^3-5x^2+6x-8x-24/x^2-4+frac{7{x}^2-13x-2}{{x}^3-{x}^2-6x}

$$\frac{ { x }^{ 2 } +x-2 }{ { x }^{ 3 } -5 { x }^{ 2 } +6x } - \frac{ 8x-24 }{ { x }^{ 2 } -4 } + \frac{ 7 { x }^{ 2 } -13x-2 }{ { x }^{ 3 } - { x }^{ 2 } -6x }$$

Evaluate

$$\frac{24}{\left(x-3\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{3}-5x^{2}+6x. Factor x^{2}-4.

$$\frac{x^{2}+x-2}{x\left(x-3\right)\left(x-2\right)}-\frac{8x-24}{\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x-2\right) and \left(x-2\right)\left(x+2\right) is x\left(x-3\right)\left(x-2\right)\left(x+2\right). Multiply \frac{x^{2}+x-2}{x\left(x-3\right)\left(x-2\right)} times \frac{x+2}{x+2}. Multiply \frac{8x-24}{\left(x-2\right)\left(x+2\right)} times \frac{x\left(x-3\right)}{x\left(x-3\right)}.

$$\frac{\left(x^{2}+x-2\right)\left(x+2\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}-\frac{\left(8x-24\right)x\left(x-3\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Since \frac{\left(x^{2}+x-2\right)\left(x+2\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)} and \frac{\left(8x-24\right)x\left(x-3\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+x-2\right)\left(x+2\right)-\left(8x-24\right)x\left(x-3\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Do the multiplications in \left(x^{2}+x-2\right)\left(x+2\right)-\left(8x-24\right)x\left(x-3\right).

$$\frac{x^{3}+2x^{2}+x^{2}+2x-2x-4-8x^{3}+24x^{2}+24x^{2}-72x}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Combine like terms in x^{3}+2x^{2}+x^{2}+2x-2x-4-8x^{3}+24x^{2}+24x^{2}-72x.

$$\frac{-7x^{3}+51x^{2}-72x-4}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Factor the expressions that are not already factored in \frac{-7x^{3}+51x^{2}-72x-4}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}.

$$\frac{\left(x-2\right)\left(-7x^{2}+37x+2\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{-7x^{2}+37x+2}{x\left(x-3\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Factor x^{3}-x^{2}-6x.

$$\frac{-7x^{2}+37x+2}{x\left(x-3\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x\left(x-3\right)\left(x+2\right)}$$

Since \frac{-7x^{2}+37x+2}{x\left(x-3\right)\left(x+2\right)} and \frac{7x^{2}-13x-2}{x\left(x-3\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}+37x+2+7x^{2}-13x-2}{x\left(x-3\right)\left(x+2\right)}$$

Combine like terms in -7x^{2}+37x+2+7x^{2}-13x-2.

$$\frac{24x}{x\left(x-3\right)\left(x+2\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{24}{\left(x-3\right)\left(x+2\right)}$$

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

$$\frac{24}{x^{2}-x-6}$$

Expand

$$\frac{24}{\left(x-3\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{3}-5x^{2}+6x. Factor x^{2}-4.

$$\frac{x^{2}+x-2}{x\left(x-3\right)\left(x-2\right)}-\frac{8x-24}{\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x-2\right) and \left(x-2\right)\left(x+2\right) is x\left(x-3\right)\left(x-2\right)\left(x+2\right). Multiply \frac{x^{2}+x-2}{x\left(x-3\right)\left(x-2\right)} times \frac{x+2}{x+2}. Multiply \frac{8x-24}{\left(x-2\right)\left(x+2\right)} times \frac{x\left(x-3\right)}{x\left(x-3\right)}.

$$\frac{\left(x^{2}+x-2\right)\left(x+2\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}-\frac{\left(8x-24\right)x\left(x-3\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Since \frac{\left(x^{2}+x-2\right)\left(x+2\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)} and \frac{\left(8x-24\right)x\left(x-3\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+x-2\right)\left(x+2\right)-\left(8x-24\right)x\left(x-3\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Do the multiplications in \left(x^{2}+x-2\right)\left(x+2\right)-\left(8x-24\right)x\left(x-3\right).

$$\frac{x^{3}+2x^{2}+x^{2}+2x-2x-4-8x^{3}+24x^{2}+24x^{2}-72x}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Combine like terms in x^{3}+2x^{2}+x^{2}+2x-2x-4-8x^{3}+24x^{2}+24x^{2}-72x.

$$\frac{-7x^{3}+51x^{2}-72x-4}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Factor the expressions that are not already factored in \frac{-7x^{3}+51x^{2}-72x-4}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}.

$$\frac{\left(x-2\right)\left(-7x^{2}+37x+2\right)}{x\left(x-3\right)\left(x-2\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{-7x^{2}+37x+2}{x\left(x-3\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x^{3}-x^{2}-6x}$$

Factor x^{3}-x^{2}-6x.

$$\frac{-7x^{2}+37x+2}{x\left(x-3\right)\left(x+2\right)}+\frac{7x^{2}-13x-2}{x\left(x-3\right)\left(x+2\right)}$$

Since \frac{-7x^{2}+37x+2}{x\left(x-3\right)\left(x+2\right)} and \frac{7x^{2}-13x-2}{x\left(x-3\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}+37x+2+7x^{2}-13x-2}{x\left(x-3\right)\left(x+2\right)}$$

Combine like terms in -7x^{2}+37x+2+7x^{2}-13x-2.

$$\frac{24x}{x\left(x-3\right)\left(x+2\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{24}{\left(x-3\right)\left(x+2\right)}$$

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

$$\frac{24}{x^{2}-x-6}$$