Solve {l}{5(x^3+1)^2}{+3(x^3+1)}

$$\left. \begin{array} { l } { 5 ( x ^ { 3 } + 1 ) ^ { 2 } } \\ { + 3 ( x ^ { 3 } + 1 ) } \end{array} \right.$$

Evaluate

$$5x^{6}+13x^{3}+8$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{3}+1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$5\left(x^{6}+2x^{3}+1\right)+3\left(x^{3}+1\right)$$

Use the distributive property to multiply 5 by x^{6}+2x^{3}+1.

$$5x^{6}+10x^{3}+5+3\left(x^{3}+1\right)$$

Use the distributive property to multiply 3 by x^{3}+1.

$$5x^{6}+10x^{3}+5+3x^{3}+3$$

Combine 10x^{3} and 3x^{3} to get 13x^{3}.

$$5x^{6}+13x^{3}+5+3$$

Add 5 and 3 to get 8.

$$5x^{6}+13x^{3}+8$$

Expand

$$5x^{6}+13x^{3}+8$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{3}+1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$5\left(x^{6}+2x^{3}+1\right)+3\left(x^{3}+1\right)$$

Use the distributive property to multiply 5 by x^{6}+2x^{3}+1.

$$5x^{6}+10x^{3}+5+3\left(x^{3}+1\right)$$

Use the distributive property to multiply 3 by x^{3}+1.

$$5x^{6}+10x^{3}+5+3x^{3}+3$$

Combine 10x^{3} and 3x^{3} to get 13x^{3}.

$$5x^{6}+13x^{3}+5+3$$

Add 5 and 3 to get 8.

$$5x^{6}+13x^{3}+8$$