Solve 2xleft(x^2+1right)^3

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

Expand

$$2x^{7}+6x^{5}+6x^{3}+2x$$

Solution Steps

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 2x by x^{6}+3x^{4}+3x^{2}+1.

$$2x^{7}+6x^{5}+6x^{3}+2x$$

Evaluate

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

Solution Steps

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 2x by x^{6}+3x^{4}+3x^{2}+1.

$$2x^{7}+6x^{5}+6x^{3}+2x$$