Solve m-2/m^2+2m-m-1/m^2+4m+4

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

Evaluate

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

Short Solution Steps

Factor m^{2}+2m. Factor m^{2}+4m+4.

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

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

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

Since \frac{\left(m-2\right)\left(m+2\right)}{m\left(m+2\right)^{2}} and \frac{\left(m-1\right)m}{m\left(m+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

Factor m^{2}+2m. Factor m^{2}+4m+4.

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

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

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

Since \frac{\left(m-2\right)\left(m+2\right)}{m\left(m+2\right)^{2}} and \frac{\left(m-1\right)m}{m\left(m+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

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