Solve a^2+a^2(a+1)^2+(a+1)^2

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

Evaluate

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

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

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

Use the distributive property to multiply a^{2} by a^{2}+2a+1.

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

Combine a^{2} and a^{2} to get 2a^{2}.

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

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

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

Combine 2a^{2} and a^{2} to get 3a^{2}.

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

Expand

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

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

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

Use the distributive property to multiply a^{2} by a^{2}+2a+1.

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

Combine a^{2} and a^{2} to get 2a^{2}.

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

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

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

Combine 2a^{2} and a^{2} to get 3a^{2}.

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