Solve (y^2+2y)^2+2(y^2+2y)+1

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

Expand

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

Solution Steps

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

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

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

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

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

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

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

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

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

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