Solve (frac{s(s^{2}+2)}{s^{2}+1})(frac{2s^{4}+4s^{3}+4s^{2}+2s+1}{2s^5+3s^4+6s^3+4s^2+2s})

$$( \frac { s ( s ^ { 2 } + 2 ) } { s ^ { 2 } + 1 } ) ( \frac { 2 s ^ { 4 } + 4 s ^ { 3 } + 4 s ^ { 2 } + 2 s + 1 } { 2 s ^ { 5 } + 3 s ^ { 4 } + 6 s ^ { 3 } + 4 s ^ { 2 } + 2 s } )$$

Evaluate

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

Short Solution Steps

Use the distributive property to multiply s by s^{2}+2.

$$\frac{s^{3}+2s}{s^{2}+1}\times \frac{2s^{4}+4s^{3}+4s^{2}+2s+1}{2s^{5}+3s^{4}+6s^{3}+4s^{2}+2s}$$

Multiply \frac{s^{3}+2s}{s^{2}+1} times \frac{2s^{4}+4s^{3}+4s^{2}+2s+1}{2s^{5}+3s^{4}+6s^{3}+4s^{2}+2s} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(s^{3}+2s\right)\left(2s^{4}+4s^{3}+4s^{2}+2s+1\right)}{\left(s^{2}+1\right)\left(2s^{5}+3s^{4}+6s^{3}+4s^{2}+2s\right)}$$

Factor the expressions that are not already factored.

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

Cancel out s in both numerator and denominator.

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

Expand the expression.

$$\frac{2s^{6}+4s^{5}+8s^{4}+10s^{3}+9s^{2}+4s+2}{2s^{6}+3s^{5}+8s^{4}+7s^{3}+8s^{2}+4s+2}$$

Expand

$$\frac{2s^{6}+4s^{5}+8s^{4}+10s^{3}+9s^{2}+4s+2}{\left(s^{2}+1\right)\left(2s^{4}+3s^{3}+6s^{2}+4s+2\right)}$$

Short Solution Steps

Use the distributive property to multiply s by s^{2}+2.

$$\frac{s^{3}+2s}{s^{2}+1}\times \frac{2s^{4}+4s^{3}+4s^{2}+2s+1}{2s^{5}+3s^{4}+6s^{3}+4s^{2}+2s}$$

Multiply \frac{s^{3}+2s}{s^{2}+1} times \frac{2s^{4}+4s^{3}+4s^{2}+2s+1}{2s^{5}+3s^{4}+6s^{3}+4s^{2}+2s} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(s^{3}+2s\right)\left(2s^{4}+4s^{3}+4s^{2}+2s+1\right)}{\left(s^{2}+1\right)\left(2s^{5}+3s^{4}+6s^{3}+4s^{2}+2s\right)}$$

Factor the expressions that are not already factored.

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

Cancel out s in both numerator and denominator.

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

Expand the expression.

$$\frac{2s^{6}+4s^{5}+8s^{4}+10s^{3}+9s^{2}+4s+2}{2s^{6}+3s^{5}+8s^{4}+7s^{3}+8s^{2}+4s+2}$$