Solve 8k/k^2+4+8k/4k+1

$$\frac { 8 k } { k ^ { 2 } + 4 } + \frac { 8 k } { 4 k + 1 }$$

Evaluate

$$\frac{8k\left(k^{2}+4k+5\right)}{\left(4k+1\right)\left(k^{2}+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of k^{2}+4 and 4k+1 is \left(4k+1\right)\left(k^{2}+4\right). Multiply \frac{8k}{k^{2}+4} times \frac{4k+1}{4k+1}. Multiply \frac{8k}{4k+1} times \frac{k^{2}+4}{k^{2}+4}.

$$\frac{8k\left(4k+1\right)}{\left(4k+1\right)\left(k^{2}+4\right)}+\frac{8k\left(k^{2}+4\right)}{\left(4k+1\right)\left(k^{2}+4\right)}$$

Since \frac{8k\left(4k+1\right)}{\left(4k+1\right)\left(k^{2}+4\right)} and \frac{8k\left(k^{2}+4\right)}{\left(4k+1\right)\left(k^{2}+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{8k\left(4k+1\right)+8k\left(k^{2}+4\right)}{\left(4k+1\right)\left(k^{2}+4\right)}$$

Do the multiplications in 8k\left(4k+1\right)+8k\left(k^{2}+4\right).

$$\frac{32k^{2}+8k+8k^{3}+32k}{\left(4k+1\right)\left(k^{2}+4\right)}$$

Combine like terms in 32k^{2}+8k+8k^{3}+32k.

$$\frac{32k^{2}+40k+8k^{3}}{\left(4k+1\right)\left(k^{2}+4\right)}$$

Expand \left(4k+1\right)\left(k^{2}+4\right).

$$\frac{32k^{2}+40k+8k^{3}}{4k^{3}+k^{2}+16k+4}$$

Factor

$$\frac{8k\left(k^{2}+4k+5\right)}{\left(4k+1\right)\left(k^{2}+4\right)}$$