Solve x^3+1+3x^2+3x/x^2+5x:(1+2/x+1/x^2)

$$\frac { x ^ { 3 } + 1 + 3 x ^ { 2 } + 3 x } { x ^ { 2 } + 5 x } : ( 1 + \frac { 2 } { x } + \frac { 1 } { x ^ { 2 } } )$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators.

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

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

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

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

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

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

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

Divide \frac{x^{3}+1+3x^{2}+3x}{x^{2}+5x} by \frac{x^{2}+2x+1}{x^{2}} by multiplying \frac{x^{3}+1+3x^{2}+3x}{x^{2}+5x} by the reciprocal of \frac{x^{2}+2x+1}{x^{2}}.

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

Factor the expressions that are not already factored.

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

Cancel out x\left(x+1\right)^{2} in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}+x}{x+5}$$

Expand

$$\frac{x^{2}+x}{x+5}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators.

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

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

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

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

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

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

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

Divide \frac{x^{3}+1+3x^{2}+3x}{x^{2}+5x} by \frac{x^{2}+2x+1}{x^{2}} by multiplying \frac{x^{3}+1+3x^{2}+3x}{x^{2}+5x} by the reciprocal of \frac{x^{2}+2x+1}{x^{2}}.

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

Factor the expressions that are not already factored.

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

Cancel out x\left(x+1\right)^{2} in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}+x}{x+5}$$