Solve {l}{(a^2+a)^2}{+8(a^2+a)}{+12}

$$\left. \begin{array} { l } { ( a ^ { 2 } + a ) ^ { 2 } } \\ { + 8 ( a ^ { 2 } + a ) } \\ { + 12 } \end{array} \right.$$

Evaluate

$$\left(a^{2}+a+2\right)\left(a^{2}+a+6\right)$$

Solution Steps

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

$$\left(a^{2}\right)^{2}+2a^{2}a+a^{2}+8\left(a^{2}+a\right)+12$$

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

$$a^{4}+2a^{2}a+a^{2}+8\left(a^{2}+a\right)+12$$

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

$$a^{4}+2a^{3}+a^{2}+8\left(a^{2}+a\right)+12$$

Use the distributive property to multiply 8 by a^{2}+a.

$$a^{4}+2a^{3}+a^{2}+8a^{2}+8a+12$$

Combine a^{2} and 8a^{2} to get 9a^{2}.

$$a^{4}+2a^{3}+9a^{2}+8a+12$$

Expand

$$a^{4}+2a^{3}+9a^{2}+8a+12$$

Solution Steps

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

$$\left(a^{2}\right)^{2}+2a^{2}a+a^{2}+8\left(a^{2}+a\right)+12$$

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

$$a^{4}+2a^{2}a+a^{2}+8\left(a^{2}+a\right)+12$$

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

$$a^{4}+2a^{3}+a^{2}+8\left(a^{2}+a\right)+12$$

Use the distributive property to multiply 8 by a^{2}+a.

$$a^{4}+2a^{3}+a^{2}+8a^{2}+8a+12$$

Combine a^{2} and 8a^{2} to get 9a^{2}.

$$a^{4}+2a^{3}+9a^{2}+8a+12$$