Solve 3x+2/x^2+3x-10-5x+11/x^2+4x-5+4x-1/x^2-3x+2

$$\frac{ 3x+2 }{ { x }^{ 2 } +3x-10 } - \frac{ 5x+11 }{ { x }^{ 2 } +4x-5 } + \frac{ 4x-1 }{ { x }^{ 2 } -3x+2 }$$

Evaluate

$$\frac{\left(x+1\right)\left(2x+15\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+3x-10. Factor x^{2}+4x-5.

$$\frac{3x+2}{\left(x-2\right)\left(x+5\right)}-\frac{5x+11}{\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+5\right) and \left(x-1\right)\left(x+5\right) is \left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{3x+2}{\left(x-2\right)\left(x+5\right)} times \frac{x-1}{x-1}. Multiply \frac{5x+11}{\left(x-1\right)\left(x+5\right)} times \frac{x-2}{x-2}.

$$\frac{\left(3x+2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}-\frac{\left(5x+11\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

Since \frac{\left(3x+2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{\left(5x+11\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+2\right)\left(x-1\right)-\left(5x+11\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

Do the multiplications in \left(3x+2\right)\left(x-1\right)-\left(5x+11\right)\left(x-2\right).

$$\frac{3x^{2}-3x+2x-2-5x^{2}+10x-11x+22}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

Combine like terms in 3x^{2}-3x+2x-2-5x^{2}+10x-11x+22.

$$\frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

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

$$\frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{\left(x-2\right)\left(x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right)\left(x+5\right) and \left(x-2\right)\left(x-1\right) is \left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{4x-1}{\left(x-2\right)\left(x-1\right)} times \frac{x+5}{x+5}.

$$\frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{\left(4x-1\right)\left(x+5\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Since \frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{\left(4x-1\right)\left(x+5\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{-2x^{2}-2x+20+\left(4x-1\right)\left(x+5\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Do the multiplications in -2x^{2}-2x+20+\left(4x-1\right)\left(x+5\right).

$$\frac{-2x^{2}-2x+20+4x^{2}+20x-x-5}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

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

$$\frac{2x^{2}+17x+15}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

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

$$\frac{2x^{2}+17x+15}{x^{3}+2x^{2}-13x+10}$$

Expand

$$\frac{2x^{2}+17x+15}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+3x-10. Factor x^{2}+4x-5.

$$\frac{3x+2}{\left(x-2\right)\left(x+5\right)}-\frac{5x+11}{\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+5\right) and \left(x-1\right)\left(x+5\right) is \left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{3x+2}{\left(x-2\right)\left(x+5\right)} times \frac{x-1}{x-1}. Multiply \frac{5x+11}{\left(x-1\right)\left(x+5\right)} times \frac{x-2}{x-2}.

$$\frac{\left(3x+2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}-\frac{\left(5x+11\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

Since \frac{\left(3x+2\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{\left(5x+11\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+2\right)\left(x-1\right)-\left(5x+11\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

Do the multiplications in \left(3x+2\right)\left(x-1\right)-\left(5x+11\right)\left(x-2\right).

$$\frac{3x^{2}-3x+2x-2-5x^{2}+10x-11x+22}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

Combine like terms in 3x^{2}-3x+2x-2-5x^{2}+10x-11x+22.

$$\frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{x^{2}-3x+2}$$

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

$$\frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{4x-1}{\left(x-2\right)\left(x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right)\left(x+5\right) and \left(x-2\right)\left(x-1\right) is \left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{4x-1}{\left(x-2\right)\left(x-1\right)} times \frac{x+5}{x+5}.

$$\frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{\left(4x-1\right)\left(x+5\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Since \frac{-2x^{2}-2x+20}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{\left(4x-1\right)\left(x+5\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{-2x^{2}-2x+20+\left(4x-1\right)\left(x+5\right)}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Do the multiplications in -2x^{2}-2x+20+\left(4x-1\right)\left(x+5\right).

$$\frac{-2x^{2}-2x+20+4x^{2}+20x-x-5}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

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

$$\frac{2x^{2}+17x+15}{\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

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

$$\frac{2x^{2}+17x+15}{x^{3}+2x^{2}-13x+10}$$