Solve ((n+1)^2-1)^2+(2n+2)^2

$$( ( n + 1 ) ^ { 2 } - 1 ) ^ { 2 } + ( 2 n + 2 ) ^ { 2 }$$

Evaluate

$$\left(n^{2}+2n+2\right)^{2}$$

Solution Steps

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

$$\left(n^{2}+2n+1-1\right)^{2}+\left(2n+2\right)^{2}$$

Subtract 1 from 1 to get 0.

$$\left(n^{2}+2n\right)^{2}+\left(2n+2\right)^{2}$$

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

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

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

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

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

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

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

$$n^{4}+4n^{3}+4n^{2}+4n^{2}+8n+4$$

Combine 4n^{2} and 4n^{2} to get 8n^{2}.

$$n^{4}+4n^{3}+8n^{2}+8n+4$$

Expand

$$n^{4}+4n^{3}+8n^{2}+8n+4$$

Solution Steps

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

$$\left(n^{2}+2n+1-1\right)^{2}+\left(2n+2\right)^{2}$$

Subtract 1 from 1 to get 0.

$$\left(n^{2}+2n\right)^{2}+\left(2n+2\right)^{2}$$

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

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

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

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

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

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

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

$$n^{4}+4n^{3}+4n^{2}+4n^{2}+8n+4$$

Combine 4n^{2} and 4n^{2} to get 8n^{2}.

$$n^{4}+4n^{3}+8n^{2}+8n+4$$