Solve x+6/x^2+x-2+3x+8/2x+4+x+1/x-1

$$\frac { x + 6 } { x ^ { 2 } + x - 2 } + \frac { 3 x + 8 } { 2 x + 4 } + \frac { x + 1 } { x - 1 }$$

Evaluate

$$\frac{\left(x+1\right)\left(5x+8\right)}{2\left(x-1\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}+x-2. Factor 2x+4.

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

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 2\left(x+2\right) is 2\left(x-1\right)\left(x+2\right). Multiply \frac{x+6}{\left(x-1\right)\left(x+2\right)} times \frac{2}{2}. Multiply \frac{3x+8}{2\left(x+2\right)} times \frac{x-1}{x-1}.

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

Since \frac{2\left(x+6\right)}{2\left(x-1\right)\left(x+2\right)} and \frac{\left(3x+8\right)\left(x-1\right)}{2\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(x+6\right)+\left(3x+8\right)\left(x-1\right).

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

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

$$\frac{7x+4+3x^{2}}{2\left(x-1\right)\left(x+2\right)}+\frac{x+1}{x-1}$$

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

$$\frac{7x+4+3x^{2}}{2\left(x-1\right)\left(x+2\right)}+\frac{\left(x+1\right)\times 2\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)}$$

Since \frac{7x+4+3x^{2}}{2\left(x-1\right)\left(x+2\right)} and \frac{\left(x+1\right)\times 2\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{7x+4+3x^{2}+\left(x+1\right)\times 2\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)}$$

Do the multiplications in 7x+4+3x^{2}+\left(x+1\right)\times 2\left(x+2\right).

$$\frac{7x+4+3x^{2}+2x^{2}+4x+2x+4}{2\left(x-1\right)\left(x+2\right)}$$

Combine like terms in 7x+4+3x^{2}+2x^{2}+4x+2x+4.

$$\frac{13x+8+5x^{2}}{2\left(x-1\right)\left(x+2\right)}$$

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

$$\frac{13x+8+5x^{2}}{2x^{2}+2x-4}$$

Expand

$$\frac{5x^{2}+13x+8}{2\left(x-1\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}+x-2. Factor 2x+4.

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

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 2\left(x+2\right) is 2\left(x-1\right)\left(x+2\right). Multiply \frac{x+6}{\left(x-1\right)\left(x+2\right)} times \frac{2}{2}. Multiply \frac{3x+8}{2\left(x+2\right)} times \frac{x-1}{x-1}.

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

Since \frac{2\left(x+6\right)}{2\left(x-1\right)\left(x+2\right)} and \frac{\left(3x+8\right)\left(x-1\right)}{2\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(x+6\right)+\left(3x+8\right)\left(x-1\right).

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

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

$$\frac{7x+4+3x^{2}}{2\left(x-1\right)\left(x+2\right)}+\frac{x+1}{x-1}$$

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

$$\frac{7x+4+3x^{2}}{2\left(x-1\right)\left(x+2\right)}+\frac{\left(x+1\right)\times 2\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)}$$

Since \frac{7x+4+3x^{2}}{2\left(x-1\right)\left(x+2\right)} and \frac{\left(x+1\right)\times 2\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{7x+4+3x^{2}+\left(x+1\right)\times 2\left(x+2\right)}{2\left(x-1\right)\left(x+2\right)}$$

Do the multiplications in 7x+4+3x^{2}+\left(x+1\right)\times 2\left(x+2\right).

$$\frac{7x+4+3x^{2}+2x^{2}+4x+2x+4}{2\left(x-1\right)\left(x+2\right)}$$

Combine like terms in 7x+4+3x^{2}+2x^{2}+4x+2x+4.

$$\frac{13x+8+5x^{2}}{2\left(x-1\right)\left(x+2\right)}$$

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

$$\frac{13x+8+5x^{2}}{2x^{2}+2x-4}$$