Solve m^2/m^2+2m+1+1/3m+3-1/6

$$\frac { m ^ { 2 } } { m ^ { 2 } + 2 m + 1 } + \frac { 1 } { 3 m + 3 } - \frac { 1 } { 6 }$$

Evaluate

$$\frac{5m^{2}+1}{6\left(m+1\right)^{2}}$$

Short Solution Steps

Factor m^{2}+2m+1. Factor 3m+3.

$$\frac{m^{2}}{\left(m+1\right)^{2}}+\frac{1}{3\left(m+1\right)}-\frac{1}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m+1\right)^{2} and 3\left(m+1\right) is 3\left(m+1\right)^{2}. Multiply \frac{m^{2}}{\left(m+1\right)^{2}} times \frac{3}{3}. Multiply \frac{1}{3\left(m+1\right)} times \frac{m+1}{m+1}.

$$\frac{3m^{2}}{3\left(m+1\right)^{2}}+\frac{m+1}{3\left(m+1\right)^{2}}-\frac{1}{6}$$

Since \frac{3m^{2}}{3\left(m+1\right)^{2}} and \frac{m+1}{3\left(m+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{3m^{2}+m+1}{3\left(m+1\right)^{2}}-\frac{1}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(m+1\right)^{2} and 6 is 6\left(m+1\right)^{2}. Multiply \frac{3m^{2}+m+1}{3\left(m+1\right)^{2}} times \frac{2}{2}. Multiply \frac{1}{6} times \frac{\left(m+1\right)^{2}}{\left(m+1\right)^{2}}.

$$\frac{2\left(3m^{2}+m+1\right)}{6\left(m+1\right)^{2}}-\frac{\left(m+1\right)^{2}}{6\left(m+1\right)^{2}}$$

Since \frac{2\left(3m^{2}+m+1\right)}{6\left(m+1\right)^{2}} and \frac{\left(m+1\right)^{2}}{6\left(m+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(3m^{2}+m+1\right)-\left(m+1\right)^{2}}{6\left(m+1\right)^{2}}$$

Do the multiplications in 2\left(3m^{2}+m+1\right)-\left(m+1\right)^{2}.

$$\frac{6m^{2}+2m+2-m^{2}-2m-1}{6\left(m+1\right)^{2}}$$

Combine like terms in 6m^{2}+2m+2-m^{2}-2m-1.

$$\frac{5m^{2}+1}{6\left(m+1\right)^{2}}$$

Expand 6\left(m+1\right)^{2}.

$$\frac{5m^{2}+1}{6m^{2}+12m+6}$$

Factor

$$\frac{5m^{2}+1}{6\left(m+1\right)^{2}}$$