Solve x^2-x/x^2+x+3x+2/x^2+x+3x+2/x+4-2x^2-4/x(x)

$$\frac { x ^ { 2 } - x } { x ^ { 2 } + x } + \frac { 3 x + 2 } { x ^ { 2 } + x } + \frac { 3 x + 2 } { x + 4 } - \frac { 2 x ^ { 2 } - 4 } { x ( x ) }$$

Evaluate

$$\frac{2x^{4}+x^{3}+8x^{2}+28x+16}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Short Solution Steps

Multiply x and x to get x^{2}.

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

Factor the expressions that are not already factored in \frac{x^{2}-x}{x^{2}+x}.

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

Cancel out x in both numerator and denominator.

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

Factor x^{2}+x.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and x+4 is x\left(x+1\right)\left(x+4\right). Multiply \frac{x^{2}+2x+2}{x\left(x+1\right)} times \frac{x+4}{x+4}. Multiply \frac{3x+2}{x+4} times \frac{x\left(x+1\right)}{x\left(x+1\right)}.

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

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

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

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

$$\frac{x^{3}+4x^{2}+2x^{2}+8x+2x+8+3x^{3}+3x^{2}+2x^{2}+2x}{x\left(x+1\right)\left(x+4\right)}-\frac{2x^{2}-4}{x^{2}}$$

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

$$\frac{4x^{3}+11x^{2}+12x+8}{x\left(x+1\right)\left(x+4\right)}-\frac{2x^{2}-4}{x^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right)\left(x+4\right) and x^{2} is \left(x+1\right)\left(x+4\right)x^{2}. Multiply \frac{4x^{3}+11x^{2}+12x+8}{x\left(x+1\right)\left(x+4\right)} times \frac{x}{x}. Multiply \frac{2x^{2}-4}{x^{2}} times \frac{\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)}.

$$\frac{\left(4x^{3}+11x^{2}+12x+8\right)x}{\left(x+1\right)\left(x+4\right)x^{2}}-\frac{\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Since \frac{\left(4x^{3}+11x^{2}+12x+8\right)x}{\left(x+1\right)\left(x+4\right)x^{2}} and \frac{\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(4x^{3}+11x^{2}+12x+8\right)x-\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Do the multiplications in \left(4x^{3}+11x^{2}+12x+8\right)x-\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right).

$$\frac{4x^{4}+11x^{3}+12x^{2}+8x-2x^{4}-10x^{3}-8x^{2}+4x^{2}+20x+16}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Combine like terms in 4x^{4}+11x^{3}+12x^{2}+8x-2x^{4}-10x^{3}-8x^{2}+4x^{2}+20x+16.

$$\frac{2x^{4}+x^{3}+8x^{2}+28x+16}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Expand \left(x+1\right)\left(x+4\right)x^{2}.

$$\frac{2x^{4}+x^{3}+8x^{2}+28x+16}{x^{4}+5x^{3}+4x^{2}}$$

Expand

$$\frac{2x^{4}+x^{3}+8x^{2}+28x+16}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Short Solution Steps

Multiply x and x to get x^{2}.

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

Factor the expressions that are not already factored in \frac{x^{2}-x}{x^{2}+x}.

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

Cancel out x in both numerator and denominator.

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

Factor x^{2}+x.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and x+4 is x\left(x+1\right)\left(x+4\right). Multiply \frac{x^{2}+2x+2}{x\left(x+1\right)} times \frac{x+4}{x+4}. Multiply \frac{3x+2}{x+4} times \frac{x\left(x+1\right)}{x\left(x+1\right)}.

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

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

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

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

$$\frac{x^{3}+4x^{2}+2x^{2}+8x+2x+8+3x^{3}+3x^{2}+2x^{2}+2x}{x\left(x+1\right)\left(x+4\right)}-\frac{2x^{2}-4}{x^{2}}$$

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

$$\frac{4x^{3}+11x^{2}+12x+8}{x\left(x+1\right)\left(x+4\right)}-\frac{2x^{2}-4}{x^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right)\left(x+4\right) and x^{2} is \left(x+1\right)\left(x+4\right)x^{2}. Multiply \frac{4x^{3}+11x^{2}+12x+8}{x\left(x+1\right)\left(x+4\right)} times \frac{x}{x}. Multiply \frac{2x^{2}-4}{x^{2}} times \frac{\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)}.

$$\frac{\left(4x^{3}+11x^{2}+12x+8\right)x}{\left(x+1\right)\left(x+4\right)x^{2}}-\frac{\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Since \frac{\left(4x^{3}+11x^{2}+12x+8\right)x}{\left(x+1\right)\left(x+4\right)x^{2}} and \frac{\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(4x^{3}+11x^{2}+12x+8\right)x-\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right)}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Do the multiplications in \left(4x^{3}+11x^{2}+12x+8\right)x-\left(2x^{2}-4\right)\left(x+1\right)\left(x+4\right).

$$\frac{4x^{4}+11x^{3}+12x^{2}+8x-2x^{4}-10x^{3}-8x^{2}+4x^{2}+20x+16}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Combine like terms in 4x^{4}+11x^{3}+12x^{2}+8x-2x^{4}-10x^{3}-8x^{2}+4x^{2}+20x+16.

$$\frac{2x^{4}+x^{3}+8x^{2}+28x+16}{\left(x+1\right)\left(x+4\right)x^{2}}$$

Expand \left(x+1\right)\left(x+4\right)x^{2}.

$$\frac{2x^{4}+x^{3}+8x^{2}+28x+16}{x^{4}+5x^{3}+4x^{2}}$$