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

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

Evaluate

$$\frac{x\left(x^{2}-6x+23\right)}{3\left(x^{2}-1\right)}$$

Short Solution Steps

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{3x-4}{x+1} times \frac{x-1}{x-1}.

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

Since \frac{x+5}{\left(x-1\right)\left(x+1\right)} and \frac{\left(3x-4\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+5-\left(3x-4\right)\left(x-1\right).

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

Combine like terms in x+5-3x^{2}+3x+4x-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and 3 is 3\left(x-1\right)\left(x+1\right). Multiply \frac{8x+1-3x^{2}}{\left(x-1\right)\left(x+1\right)} times \frac{3}{3}. Multiply \frac{x+3}{3} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

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

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

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

$$\frac{24x+3-9x^{2}+x^{3}-x+3x^{2}-3}{3\left(x-1\right)\left(x+1\right)}$$

Combine like terms in 24x+3-9x^{2}+x^{3}-x+3x^{2}-3.

$$\frac{23x-6x^{2}+x^{3}}{3\left(x-1\right)\left(x+1\right)}$$

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

$$\frac{23x-6x^{2}+x^{3}}{3x^{2}-3}$$

Expand

$$\frac{x^{3}-6x^{2}+23x}{3\left(x^{2}-1\right)}$$

Short Solution Steps

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{3x-4}{x+1} times \frac{x-1}{x-1}.

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

Since \frac{x+5}{\left(x-1\right)\left(x+1\right)} and \frac{\left(3x-4\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+5-\left(3x-4\right)\left(x-1\right).

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

Combine like terms in x+5-3x^{2}+3x+4x-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and 3 is 3\left(x-1\right)\left(x+1\right). Multiply \frac{8x+1-3x^{2}}{\left(x-1\right)\left(x+1\right)} times \frac{3}{3}. Multiply \frac{x+3}{3} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

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

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

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

$$\frac{24x+3-9x^{2}+x^{3}-x+3x^{2}-3}{3\left(x-1\right)\left(x+1\right)}$$

Combine like terms in 24x+3-9x^{2}+x^{3}-x+3x^{2}-3.

$$\frac{23x-6x^{2}+x^{3}}{3\left(x-1\right)\left(x+1\right)}$$

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

$$\frac{23x-6x^{2}+x^{3}}{3x^{2}-3}$$