Solve [1/x-20/x+2+3/x+3]divx/x+2+frac{x}{x+3}=

$$[ \frac { 1 } { x } - \frac { 20 } { x + 2 } + \frac { 3 } { x + 3 } ] \div \frac { x } { x + 2 } + \frac { x } { x + 3 } =$$

Evaluate

$$\frac{x^{3}-16x^{2}-49x+6}{\left(x+3\right)x^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+2 is x\left(x+2\right). Multiply \frac{1}{x} times \frac{x+2}{x+2}. Multiply \frac{20}{x+2} times \frac{x}{x}.

$$\frac{\frac{x+2}{x\left(x+2\right)}-\frac{20x}{x\left(x+2\right)}+\frac{3}{x+3}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Since \frac{x+2}{x\left(x+2\right)} and \frac{20x}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x+2-20x}{x\left(x+2\right)}+\frac{3}{x+3}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Combine like terms in x+2-20x.

$$\frac{\frac{-19x+2}{x\left(x+2\right)}+\frac{3}{x+3}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and x+3 is x\left(x+2\right)\left(x+3\right). Multiply \frac{-19x+2}{x\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{3}{x+3} times \frac{x\left(x+2\right)}{x\left(x+2\right)}.

$$\frac{\frac{\left(-19x+2\right)\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Since \frac{\left(-19x+2\right)\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)} and \frac{3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(-19x+2\right)\left(x+3\right)+3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Do the multiplications in \left(-19x+2\right)\left(x+3\right)+3x\left(x+2\right).

$$\frac{\frac{-19x^{2}-57x+2x+6+3x^{2}+6x}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Combine like terms in -19x^{2}-57x+2x+6+3x^{2}+6x.

$$\frac{\frac{-16x^{2}-49x+6}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Divide \frac{-16x^{2}-49x+6}{x\left(x+2\right)\left(x+3\right)} by \frac{x}{x+2} by multiplying \frac{-16x^{2}-49x+6}{x\left(x+2\right)\left(x+3\right)} by the reciprocal of \frac{x}{x+2}.

$$\frac{\left(-16x^{2}-49x+6\right)\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)x}+\frac{x}{x+3}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{-16x^{2}-49x+6}{xx\left(x+3\right)}+\frac{x}{x+3}$$

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

$$\frac{-16x^{2}-49x+6}{x^{2}\left(x+3\right)}+\frac{x}{x+3}$$

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

$$\frac{-16x^{2}-49x+6}{\left(x+3\right)x^{2}}+\frac{xx^{2}}{\left(x+3\right)x^{2}}$$

Since \frac{-16x^{2}-49x+6}{\left(x+3\right)x^{2}} and \frac{xx^{2}}{\left(x+3\right)x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-16x^{2}-49x+6+xx^{2}}{\left(x+3\right)x^{2}}$$

Do the multiplications in -16x^{2}-49x+6+xx^{2}.

$$\frac{-16x^{2}-49x+6+x^{3}}{\left(x+3\right)x^{2}}$$

Expand \left(x+3\right)x^{2}.

$$\frac{-16x^{2}-49x+6+x^{3}}{x^{3}+3x^{2}}$$

Expand

$$\frac{x^{3}-16x^{2}-49x+6}{\left(x+3\right)x^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+2 is x\left(x+2\right). Multiply \frac{1}{x} times \frac{x+2}{x+2}. Multiply \frac{20}{x+2} times \frac{x}{x}.

$$\frac{\frac{x+2}{x\left(x+2\right)}-\frac{20x}{x\left(x+2\right)}+\frac{3}{x+3}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Since \frac{x+2}{x\left(x+2\right)} and \frac{20x}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x+2-20x}{x\left(x+2\right)}+\frac{3}{x+3}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Combine like terms in x+2-20x.

$$\frac{\frac{-19x+2}{x\left(x+2\right)}+\frac{3}{x+3}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and x+3 is x\left(x+2\right)\left(x+3\right). Multiply \frac{-19x+2}{x\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{3}{x+3} times \frac{x\left(x+2\right)}{x\left(x+2\right)}.

$$\frac{\frac{\left(-19x+2\right)\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Since \frac{\left(-19x+2\right)\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)} and \frac{3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(-19x+2\right)\left(x+3\right)+3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Do the multiplications in \left(-19x+2\right)\left(x+3\right)+3x\left(x+2\right).

$$\frac{\frac{-19x^{2}-57x+2x+6+3x^{2}+6x}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Combine like terms in -19x^{2}-57x+2x+6+3x^{2}+6x.

$$\frac{\frac{-16x^{2}-49x+6}{x\left(x+2\right)\left(x+3\right)}}{\frac{x}{x+2}}+\frac{x}{x+3}$$

Divide \frac{-16x^{2}-49x+6}{x\left(x+2\right)\left(x+3\right)} by \frac{x}{x+2} by multiplying \frac{-16x^{2}-49x+6}{x\left(x+2\right)\left(x+3\right)} by the reciprocal of \frac{x}{x+2}.

$$\frac{\left(-16x^{2}-49x+6\right)\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)x}+\frac{x}{x+3}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{-16x^{2}-49x+6}{xx\left(x+3\right)}+\frac{x}{x+3}$$

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

$$\frac{-16x^{2}-49x+6}{x^{2}\left(x+3\right)}+\frac{x}{x+3}$$

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

$$\frac{-16x^{2}-49x+6}{\left(x+3\right)x^{2}}+\frac{xx^{2}}{\left(x+3\right)x^{2}}$$

Since \frac{-16x^{2}-49x+6}{\left(x+3\right)x^{2}} and \frac{xx^{2}}{\left(x+3\right)x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-16x^{2}-49x+6+xx^{2}}{\left(x+3\right)x^{2}}$$

Do the multiplications in -16x^{2}-49x+6+xx^{2}.

$$\frac{-16x^{2}-49x+6+x^{3}}{\left(x+3\right)x^{2}}$$

Expand \left(x+3\right)x^{2}.

$$\frac{-16x^{2}-49x+6+x^{3}}{x^{3}+3x^{2}}$$