Solve 3m/4m-3+m^2/4m^2+7m+3

$$\frac { 3 m } { 4 m - 3 } + \frac { m ^ { 2 } } { 4 m ^ { 2 } + 7 m + 3 }$$

Evaluate

$$\frac{m\left(16m^{2}+18m+9\right)}{\left(m+1\right)\left(16m^{2}-9\right)}$$

Short Solution Steps

Factor 4m^{2}+7m+3.

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

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

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

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

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

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

$$\frac{12m^{3}+9m^{2}+12m^{2}+9m+4m^{3}-3m^{2}}{\left(4m-3\right)\left(m+1\right)\left(4m+3\right)}$$

Combine like terms in 12m^{3}+9m^{2}+12m^{2}+9m+4m^{3}-3m^{2}.

$$\frac{16m^{3}+18m^{2}+9m}{\left(4m-3\right)\left(m+1\right)\left(4m+3\right)}$$

Expand \left(4m-3\right)\left(m+1\right)\left(4m+3\right).

$$\frac{16m^{3}+18m^{2}+9m}{16m^{3}+16m^{2}-9m-9}$$

Factor

$$\frac{m\left(16m^{2}+18m+9\right)}{\left(m+1\right)\left(16m^{2}-9\right)}$$