Solve m+4/m^2+2m-8-4/m^2+8m+16-m-4/m^2+2m-8

$$\frac { m + 4 } { m ^ { 2 } + 2 m - 8 } - \frac { 4 } { m ^ { 2 } + 8 m + 16 } - \frac { m - 4 } { m ^ { 2 } + 2 m - 8 }$$

Evaluate

$$\frac{4\left(m+10\right)}{\left(m-2\right)\left(m+4\right)^{2}}$$

Short Solution Steps

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

$$\frac{m+4}{\left(m-2\right)\left(m+4\right)}-\frac{4}{m^{2}+8m+16}-\frac{m-4}{m^{2}+2m-8}$$

Cancel out m+4 in both numerator and denominator.

$$\frac{1}{m-2}-\frac{4}{m^{2}+8m+16}-\frac{m-4}{m^{2}+2m-8}$$

Factor m^{2}+8m+16.

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

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

$$\frac{\left(m+4\right)^{2}}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{4\left(m-2\right)}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Since \frac{\left(m+4\right)^{2}}{\left(m-2\right)\left(m+4\right)^{2}} and \frac{4\left(m-2\right)}{\left(m-2\right)\left(m+4\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(m+4\right)^{2}-4\left(m-2\right)}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Do the multiplications in \left(m+4\right)^{2}-4\left(m-2\right).

$$\frac{m^{2}+8m+16-4m+8}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Combine like terms in m^{2}+8m+16-4m+8.

$$\frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Factor m^{2}+2m-8.

$$\frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{\left(m-2\right)\left(m+4\right)}$$

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

$$\frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{\left(m-4\right)\left(m+4\right)}{\left(m-2\right)\left(m+4\right)^{2}}$$

Since \frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}} and \frac{\left(m-4\right)\left(m+4\right)}{\left(m-2\right)\left(m+4\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{m^{2}+4m+24-\left(m-4\right)\left(m+4\right)}{\left(m-2\right)\left(m+4\right)^{2}}$$

Do the multiplications in m^{2}+4m+24-\left(m-4\right)\left(m+4\right).

$$\frac{m^{2}+4m+24-m^{2}-4m+4m+16}{\left(m-2\right)\left(m+4\right)^{2}}$$

Combine like terms in m^{2}+4m+24-m^{2}-4m+4m+16.

$$\frac{4m+40}{\left(m-2\right)\left(m+4\right)^{2}}$$

Expand \left(m-2\right)\left(m+4\right)^{2}.

$$\frac{4m+40}{m^{3}+6m^{2}-32}$$

Expand

$$\frac{4\left(m+10\right)}{\left(m-2\right)\left(m+4\right)^{2}}$$

Short Solution Steps

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

$$\frac{m+4}{\left(m-2\right)\left(m+4\right)}-\frac{4}{m^{2}+8m+16}-\frac{m-4}{m^{2}+2m-8}$$

Cancel out m+4 in both numerator and denominator.

$$\frac{1}{m-2}-\frac{4}{m^{2}+8m+16}-\frac{m-4}{m^{2}+2m-8}$$

Factor m^{2}+8m+16.

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

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

$$\frac{\left(m+4\right)^{2}}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{4\left(m-2\right)}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Since \frac{\left(m+4\right)^{2}}{\left(m-2\right)\left(m+4\right)^{2}} and \frac{4\left(m-2\right)}{\left(m-2\right)\left(m+4\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(m+4\right)^{2}-4\left(m-2\right)}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Do the multiplications in \left(m+4\right)^{2}-4\left(m-2\right).

$$\frac{m^{2}+8m+16-4m+8}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Combine like terms in m^{2}+8m+16-4m+8.

$$\frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{m^{2}+2m-8}$$

Factor m^{2}+2m-8.

$$\frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{m-4}{\left(m-2\right)\left(m+4\right)}$$

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

$$\frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}}-\frac{\left(m-4\right)\left(m+4\right)}{\left(m-2\right)\left(m+4\right)^{2}}$$

Since \frac{m^{2}+4m+24}{\left(m-2\right)\left(m+4\right)^{2}} and \frac{\left(m-4\right)\left(m+4\right)}{\left(m-2\right)\left(m+4\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{m^{2}+4m+24-\left(m-4\right)\left(m+4\right)}{\left(m-2\right)\left(m+4\right)^{2}}$$

Do the multiplications in m^{2}+4m+24-\left(m-4\right)\left(m+4\right).

$$\frac{m^{2}+4m+24-m^{2}-4m+4m+16}{\left(m-2\right)\left(m+4\right)^{2}}$$

Combine like terms in m^{2}+4m+24-m^{2}-4m+4m+16.

$$\frac{4m+40}{\left(m-2\right)\left(m+4\right)^{2}}$$

Expand \left(m-2\right)\left(m+4\right)^{2}.

$$\frac{4m+40}{m^{3}+6m^{2}-32}$$