Solve x+4/x^2+6x+8+6/(x+4)-10/(x+2)

$$\frac { x + 4 } { x ^ { 2 } + 6 x + 8 } + \frac { 6 } { ( x + 4 ) } - \frac { 10 } { ( x + 2 ) }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x+4}{x^{2}+6x+8}.

$$\frac{x+4}{\left(x+2\right)\left(x+4\right)}+\frac{6}{x+4}-\frac{10}{x+2}$$

Cancel out x+4 in both numerator and denominator.

$$\frac{1}{x+2}+\frac{6}{x+4}-\frac{10}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x+4 is \left(x+2\right)\left(x+4\right). Multiply \frac{1}{x+2} times \frac{x+4}{x+4}. Multiply \frac{6}{x+4} times \frac{x+2}{x+2}.

$$\frac{x+4}{\left(x+2\right)\left(x+4\right)}+\frac{6\left(x+2\right)}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

Since \frac{x+4}{\left(x+2\right)\left(x+4\right)} and \frac{6\left(x+2\right)}{\left(x+2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+4+6\left(x+2\right)}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

Do the multiplications in x+4+6\left(x+2\right).

$$\frac{x+4+6x+12}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

Combine like terms in x+4+6x+12.

$$\frac{7x+16}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x+4\right) and x+2 is \left(x+2\right)\left(x+4\right). Multiply \frac{10}{x+2} times \frac{x+4}{x+4}.

$$\frac{7x+16}{\left(x+2\right)\left(x+4\right)}-\frac{10\left(x+4\right)}{\left(x+2\right)\left(x+4\right)}$$

Since \frac{7x+16}{\left(x+2\right)\left(x+4\right)} and \frac{10\left(x+4\right)}{\left(x+2\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x+16-10\left(x+4\right)}{\left(x+2\right)\left(x+4\right)}$$

Do the multiplications in 7x+16-10\left(x+4\right).

$$\frac{7x+16-10x-40}{\left(x+2\right)\left(x+4\right)}$$

Combine like terms in 7x+16-10x-40.

$$\frac{-3x-24}{\left(x+2\right)\left(x+4\right)}$$

Expand \left(x+2\right)\left(x+4\right).

$$\frac{-3x-24}{x^{2}+6x+8}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x+4}{x^{2}+6x+8}.

$$\frac{x+4}{\left(x+2\right)\left(x+4\right)}+\frac{6}{x+4}-\frac{10}{x+2}$$

Cancel out x+4 in both numerator and denominator.

$$\frac{1}{x+2}+\frac{6}{x+4}-\frac{10}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x+4 is \left(x+2\right)\left(x+4\right). Multiply \frac{1}{x+2} times \frac{x+4}{x+4}. Multiply \frac{6}{x+4} times \frac{x+2}{x+2}.

$$\frac{x+4}{\left(x+2\right)\left(x+4\right)}+\frac{6\left(x+2\right)}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

Since \frac{x+4}{\left(x+2\right)\left(x+4\right)} and \frac{6\left(x+2\right)}{\left(x+2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+4+6\left(x+2\right)}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

Do the multiplications in x+4+6\left(x+2\right).

$$\frac{x+4+6x+12}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

Combine like terms in x+4+6x+12.

$$\frac{7x+16}{\left(x+2\right)\left(x+4\right)}-\frac{10}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x+4\right) and x+2 is \left(x+2\right)\left(x+4\right). Multiply \frac{10}{x+2} times \frac{x+4}{x+4}.

$$\frac{7x+16}{\left(x+2\right)\left(x+4\right)}-\frac{10\left(x+4\right)}{\left(x+2\right)\left(x+4\right)}$$

Since \frac{7x+16}{\left(x+2\right)\left(x+4\right)} and \frac{10\left(x+4\right)}{\left(x+2\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x+16-10\left(x+4\right)}{\left(x+2\right)\left(x+4\right)}$$

Do the multiplications in 7x+16-10\left(x+4\right).

$$\frac{7x+16-10x-40}{\left(x+2\right)\left(x+4\right)}$$

Combine like terms in 7x+16-10x-40.

$$\frac{-3x-24}{\left(x+2\right)\left(x+4\right)}$$

Expand \left(x+2\right)\left(x+4\right).

$$\frac{-3x-24}{x^{2}+6x+8}$$