Solve frac{x^{2}-x-12}{x^{2}+x-2}+x^2-6x+8/x^2-3x-10+x^2-3x+2/x^2-2x-15

$$\frac { x ^ { 2 } - x - 12 } { x ^ { 2 } + x - 2 } + \frac { x ^ { 2 } - 6 x + 8 } { x ^ { 2 } - 3 x - 10 } + \frac { x ^ { 2 } - 3 x + 2 } { x ^ { 2 } - 2 x - 15 }$$

Evaluate

$$\frac{3x^{4}-9x^{3}-35x^{2}+81x+152}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+x-2. Factor x^{2}-3x-10.

$$\frac{x^{2}-x-12}{\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-6x+8}{\left(x-5\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+2\right) and \left(x-5\right)\left(x+2\right) is \left(x-5\right)\left(x-1\right)\left(x+2\right). Multiply \frac{x^{2}-x-12}{\left(x-1\right)\left(x+2\right)} times \frac{x-5}{x-5}. Multiply \frac{x^{2}-6x+8}{\left(x-5\right)\left(x+2\right)} times \frac{x-1}{x-1}.

$$\frac{\left(x^{2}-x-12\right)\left(x-5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{\left(x^{2}-6x+8\right)\left(x-1\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Since \frac{\left(x^{2}-x-12\right)\left(x-5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} and \frac{\left(x^{2}-6x+8\right)\left(x-1\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}-x-12\right)\left(x-5\right)+\left(x^{2}-6x+8\right)\left(x-1\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Do the multiplications in \left(x^{2}-x-12\right)\left(x-5\right)+\left(x^{2}-6x+8\right)\left(x-1\right).

$$\frac{x^{3}-5x^{2}-x^{2}+5x-12x+60+x^{3}-x^{2}-6x^{2}+6x+8x-8}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Combine like terms in x^{3}-5x^{2}-x^{2}+5x-12x+60+x^{3}-x^{2}-6x^{2}+6x+8x-8.

$$\frac{2x^{3}-13x^{2}+7x+52}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Factor x^{2}-2x-15.

$$\frac{2x^{3}-13x^{2}+7x+52}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{\left(x-5\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-1\right)\left(x+2\right) and \left(x-5\right)\left(x+3\right) is \left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right). Multiply \frac{2x^{3}-13x^{2}+7x+52}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{x^{2}-3x+2}{\left(x-5\right)\left(x+3\right)} times \frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}.

$$\frac{\left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}+\frac{\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Since \frac{\left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} and \frac{\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)+\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in \left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)+\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right).

$$\frac{2x^{4}+6x^{3}-13x^{3}-39x^{2}+7x^{2}+21x+52x+156+x^{4}+x^{3}-2x^{2}-3x^{3}-3x^{2}+6x+2x^{2}+2x-4}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 2x^{4}+6x^{3}-13x^{3}-39x^{2}+7x^{2}+21x+52x+156+x^{4}+x^{3}-2x^{2}-3x^{3}-3x^{2}+6x+2x^{2}+2x-4.

$$\frac{3x^{4}-9x^{3}-35x^{2}+81x+152}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Expand \left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right).

$$\frac{3x^{4}-9x^{3}-35x^{2}+81x+152}{x^{4}-x^{3}-19x^{2}-11x+30}$$

Expand

$$\frac{3x^{4}-9x^{3}-35x^{2}+81x+152}{\left(x-1\right)\left(x+2\right)\left(x^{2}-2x-15\right)}$$

Short Solution Steps

Factor x^{2}+x-2. Factor x^{2}-3x-10.

$$\frac{x^{2}-x-12}{\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-6x+8}{\left(x-5\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+2\right) and \left(x-5\right)\left(x+2\right) is \left(x-5\right)\left(x-1\right)\left(x+2\right). Multiply \frac{x^{2}-x-12}{\left(x-1\right)\left(x+2\right)} times \frac{x-5}{x-5}. Multiply \frac{x^{2}-6x+8}{\left(x-5\right)\left(x+2\right)} times \frac{x-1}{x-1}.

$$\frac{\left(x^{2}-x-12\right)\left(x-5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{\left(x^{2}-6x+8\right)\left(x-1\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Since \frac{\left(x^{2}-x-12\right)\left(x-5\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} and \frac{\left(x^{2}-6x+8\right)\left(x-1\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}-x-12\right)\left(x-5\right)+\left(x^{2}-6x+8\right)\left(x-1\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Do the multiplications in \left(x^{2}-x-12\right)\left(x-5\right)+\left(x^{2}-6x+8\right)\left(x-1\right).

$$\frac{x^{3}-5x^{2}-x^{2}+5x-12x+60+x^{3}-x^{2}-6x^{2}+6x+8x-8}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Combine like terms in x^{3}-5x^{2}-x^{2}+5x-12x+60+x^{3}-x^{2}-6x^{2}+6x+8x-8.

$$\frac{2x^{3}-13x^{2}+7x+52}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{x^{2}-2x-15}$$

Factor x^{2}-2x-15.

$$\frac{2x^{3}-13x^{2}+7x+52}{\left(x-5\right)\left(x-1\right)\left(x+2\right)}+\frac{x^{2}-3x+2}{\left(x-5\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-1\right)\left(x+2\right) and \left(x-5\right)\left(x+3\right) is \left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right). Multiply \frac{2x^{3}-13x^{2}+7x+52}{\left(x-5\right)\left(x-1\right)\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{x^{2}-3x+2}{\left(x-5\right)\left(x+3\right)} times \frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}.

$$\frac{\left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}+\frac{\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Since \frac{\left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} and \frac{\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)+\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in \left(2x^{3}-13x^{2}+7x+52\right)\left(x+3\right)+\left(x^{2}-3x+2\right)\left(x-1\right)\left(x+2\right).

$$\frac{2x^{4}+6x^{3}-13x^{3}-39x^{2}+7x^{2}+21x+52x+156+x^{4}+x^{3}-2x^{2}-3x^{3}-3x^{2}+6x+2x^{2}+2x-4}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 2x^{4}+6x^{3}-13x^{3}-39x^{2}+7x^{2}+21x+52x+156+x^{4}+x^{3}-2x^{2}-3x^{3}-3x^{2}+6x+2x^{2}+2x-4.

$$\frac{3x^{4}-9x^{3}-35x^{2}+81x+152}{\left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Expand \left(x-5\right)\left(x-1\right)\left(x+2\right)\left(x+3\right).

$$\frac{3x^{4}-9x^{3}-35x^{2}+81x+152}{x^{4}-x^{3}-19x^{2}-11x+30}$$