Solve (x^2+2x)^2+2(x^2+2x)+1

$$( x ^ { 2 } + 2 x ) ^ { 2 } + 2 ( x ^ { 2 } + 2 x ) + 1$$

Evaluate

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

Solution Steps

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

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

$$x^{4}+4x^{3}+4x^{2}+2x^{2}+4x+1$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$x^{4}+4x^{3}+6x^{2}+4x+1$$

Expand

$$x^{4}+4x^{3}+6x^{2}+4x+1$$

Solution Steps

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

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

$$x^{4}+4x^{3}+4x^{2}+2x^{2}+4x+1$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$x^{4}+4x^{3}+6x^{2}+4x+1$$