Solve m^2+1/m+12+2/m

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

Evaluate

$$\frac{m^{4}+12m^{3}+3m+24}{m\left(m+12\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply m^{2} times \frac{m+12}{m+12}.

$$\frac{m^{2}\left(m+12\right)}{m+12}+\frac{1}{m+12}+\frac{2}{m}$$

Since \frac{m^{2}\left(m+12\right)}{m+12} and \frac{1}{m+12} have the same denominator, add them by adding their numerators.

$$\frac{m^{2}\left(m+12\right)+1}{m+12}+\frac{2}{m}$$

Do the multiplications in m^{2}\left(m+12\right)+1.

$$\frac{m^{3}+12m^{2}+1}{m+12}+\frac{2}{m}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m+12 and m is m\left(m+12\right). Multiply \frac{m^{3}+12m^{2}+1}{m+12} times \frac{m}{m}. Multiply \frac{2}{m} times \frac{m+12}{m+12}.

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

Since \frac{\left(m^{3}+12m^{2}+1\right)m}{m\left(m+12\right)} and \frac{2\left(m+12\right)}{m\left(m+12\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(m^{3}+12m^{2}+1\right)m+2\left(m+12\right).

$$\frac{m^{4}+12m^{3}+m+2m+24}{m\left(m+12\right)}$$

Combine like terms in m^{4}+12m^{3}+m+2m+24.

$$\frac{m^{4}+12m^{3}+3m+24}{m\left(m+12\right)}$$

Expand m\left(m+12\right).

$$\frac{m^{4}+12m^{3}+3m+24}{m^{2}+12m}$$

Differentiate w.r.t. m

$$\frac{2m^{5}+48m^{4}+288m^{3}-3m^{2}-48m-288}{\left(m\left(m+12\right)\right)^{2}}$$