Solve s/2s+1+s/s^2+1

$$\frac { s } { 2 s + 1 } + \frac { s } { s ^ { 2 } + 1 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in s^{3}+s+2s^{2}+s.

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

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

$$\frac{s^{3}+2s+2s^{2}}{2s^{3}+s^{2}+2s+1}$$

Factor

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