Solve n^2-4m+3/m^2-1+m^2+6m/3m^2+21m+18

$$\frac { n ^ { 2 } - 4 m + 3 } { m ^ { 2 } - 1 } + \frac { m ^ { 2 } + 6 m } { 3 m ^ { 2 } + 21 m + 18 }$$

Evaluate

$$\frac{m^{2}-13m+3n^{2}+9}{3\left(m^{2}-1\right)}$$

$$m\neq -6\text{ and }|m|\neq 1$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{m^{2}+6m}{3m^{2}+21m+18}.

$$\frac{n^{2}-4m+3}{m^{2}-1}+\frac{m\left(m+6\right)}{3\left(m+1\right)\left(m+6\right)}$$

Cancel out m+6 in both numerator and denominator.

$$\frac{n^{2}-4m+3}{m^{2}-1}+\frac{m}{3\left(m+1\right)}$$

Factor m^{2}-1.

$$\frac{n^{2}-4m+3}{\left(m-1\right)\left(m+1\right)}+\frac{m}{3\left(m+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m-1\right)\left(m+1\right) and 3\left(m+1\right) is 3\left(m-1\right)\left(m+1\right). Multiply \frac{n^{2}-4m+3}{\left(m-1\right)\left(m+1\right)} times \frac{3}{3}. Multiply \frac{m}{3\left(m+1\right)} times \frac{m-1}{m-1}.

$$\frac{3\left(n^{2}-4m+3\right)}{3\left(m-1\right)\left(m+1\right)}+\frac{m\left(m-1\right)}{3\left(m-1\right)\left(m+1\right)}$$

Since \frac{3\left(n^{2}-4m+3\right)}{3\left(m-1\right)\left(m+1\right)} and \frac{m\left(m-1\right)}{3\left(m-1\right)\left(m+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(n^{2}-4m+3\right)+m\left(m-1\right)}{3\left(m-1\right)\left(m+1\right)}$$

Do the multiplications in 3\left(n^{2}-4m+3\right)+m\left(m-1\right).

$$\frac{3n^{2}-12m+9+m^{2}-m}{3\left(m-1\right)\left(m+1\right)}$$

Combine like terms in 3n^{2}-12m+9+m^{2}-m.

$$\frac{3n^{2}-13m+9+m^{2}}{3\left(m-1\right)\left(m+1\right)}$$

Expand 3\left(m-1\right)\left(m+1\right).

$$\frac{3n^{2}-13m+9+m^{2}}{3m^{2}-3}$$

Expand

$$\frac{m^{2}-13m+3n^{2}+9}{3\left(m^{2}-1\right)}$$

$$m\neq -6\text{ and }|m|\neq 1$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{m^{2}+6m}{3m^{2}+21m+18}.

$$\frac{n^{2}-4m+3}{m^{2}-1}+\frac{m\left(m+6\right)}{3\left(m+1\right)\left(m+6\right)}$$

Cancel out m+6 in both numerator and denominator.

$$\frac{n^{2}-4m+3}{m^{2}-1}+\frac{m}{3\left(m+1\right)}$$

Factor m^{2}-1.

$$\frac{n^{2}-4m+3}{\left(m-1\right)\left(m+1\right)}+\frac{m}{3\left(m+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m-1\right)\left(m+1\right) and 3\left(m+1\right) is 3\left(m-1\right)\left(m+1\right). Multiply \frac{n^{2}-4m+3}{\left(m-1\right)\left(m+1\right)} times \frac{3}{3}. Multiply \frac{m}{3\left(m+1\right)} times \frac{m-1}{m-1}.

$$\frac{3\left(n^{2}-4m+3\right)}{3\left(m-1\right)\left(m+1\right)}+\frac{m\left(m-1\right)}{3\left(m-1\right)\left(m+1\right)}$$

Since \frac{3\left(n^{2}-4m+3\right)}{3\left(m-1\right)\left(m+1\right)} and \frac{m\left(m-1\right)}{3\left(m-1\right)\left(m+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(n^{2}-4m+3\right)+m\left(m-1\right)}{3\left(m-1\right)\left(m+1\right)}$$

Do the multiplications in 3\left(n^{2}-4m+3\right)+m\left(m-1\right).

$$\frac{3n^{2}-12m+9+m^{2}-m}{3\left(m-1\right)\left(m+1\right)}$$

Combine like terms in 3n^{2}-12m+9+m^{2}-m.

$$\frac{3n^{2}-13m+9+m^{2}}{3\left(m-1\right)\left(m+1\right)}$$

Expand 3\left(m-1\right)\left(m+1\right).

$$\frac{3n^{2}-13m+9+m^{2}}{3m^{2}-3}$$