Solve (14x)(5/2)(2x^2+2)^3/2(4x)+(2x^2+2)^5/2(14)

$$( 14 x ) ( \frac { 5 } { 2 } ) ( 2 x ^ { 2 } + 2 ) ^ { 3 / 2 } ( 4 x ) + ( 2 x ^ { 2 } + 2 ) ^ { 5 / 2 } ( 14 )$$

Factor

$$28\left(6x^{2}+1\right)\left(2x^{2}+2\right)^{\frac{3}{2}}$$

Solution Steps

Factor out 56.

$$56\left(5x\sqrt{2}\left(x^{2}+1\right)^{\frac{3}{2}}x+\sqrt{2}\left(x^{2}+1\right)^{\frac{5}{2}}\right)$$

Consider 5\sqrt{2}\left(x^{2}+1\right)^{\frac{3}{2}}x^{2}+\sqrt{2}\left(x^{2}+1\right)^{\frac{5}{2}}. Factor out \sqrt{2}\left(x^{2}+1\right)^{\frac{3}{2}}.

$$\sqrt{2}\left(x^{2}+1\right)^{\frac{3}{2}}\left(5x^{2}+x^{2}+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: 6x^{2}+1,x^{2}+1.

$$56\sqrt{2}\left(x^{2}+1\right)^{\frac{3}{2}}\left(6x^{2}+1\right)$$

Evaluate

$$56\sqrt{2}\left(6x^{2}+1\right)\left(x^{2}+1\right)^{\frac{3}{2}}$$