Solve (2a+3b)^2+2(2a+3b)(2a-3b)+(2a-3b)^2

$$( 2 a + 3 b ) ^ { 2 } + 2 ( 2 a + 3 b ) ( 2 a - 3 b ) + ( 2 a - 3 b ) ^ { 2 }$$

Evaluate

$$16a^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a+3b\right)^{2}.

$$4a^{2}+12ab+9b^{2}+2\left(2a+3b\right)\left(2a-3b\right)+\left(2a-3b\right)^{2}$$

Use the distributive property to multiply 2 by 2a+3b.

$$4a^{2}+12ab+9b^{2}+\left(4a+6b\right)\left(2a-3b\right)+\left(2a-3b\right)^{2}$$

Use the distributive property to multiply 4a+6b by 2a-3b and combine like terms.

$$4a^{2}+12ab+9b^{2}+8a^{2}-18b^{2}+\left(2a-3b\right)^{2}$$

Combine 4a^{2} and 8a^{2} to get 12a^{2}.

$$12a^{2}+12ab+9b^{2}-18b^{2}+\left(2a-3b\right)^{2}$$

Combine 9b^{2} and -18b^{2} to get -9b^{2}.

$$12a^{2}+12ab-9b^{2}+\left(2a-3b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a-3b\right)^{2}.

$$12a^{2}+12ab-9b^{2}+4a^{2}-12ab+9b^{2}$$

Combine 12a^{2} and 4a^{2} to get 16a^{2}.

$$16a^{2}+12ab-9b^{2}-12ab+9b^{2}$$

Combine 12ab and -12ab to get 0.

$$16a^{2}-9b^{2}+9b^{2}$$

Combine -9b^{2} and 9b^{2} to get 0.

$$16a^{2}$$

Expand

$$16a^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a+3b\right)^{2}.

$$4a^{2}+12ab+9b^{2}+2\left(2a+3b\right)\left(2a-3b\right)+\left(2a-3b\right)^{2}$$

Use the distributive property to multiply 2 by 2a+3b.

$$4a^{2}+12ab+9b^{2}+\left(4a+6b\right)\left(2a-3b\right)+\left(2a-3b\right)^{2}$$

Use the distributive property to multiply 4a+6b by 2a-3b and combine like terms.

$$4a^{2}+12ab+9b^{2}+8a^{2}-18b^{2}+\left(2a-3b\right)^{2}$$

Combine 4a^{2} and 8a^{2} to get 12a^{2}.

$$12a^{2}+12ab+9b^{2}-18b^{2}+\left(2a-3b\right)^{2}$$

Combine 9b^{2} and -18b^{2} to get -9b^{2}.

$$12a^{2}+12ab-9b^{2}+\left(2a-3b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a-3b\right)^{2}.

$$12a^{2}+12ab-9b^{2}+4a^{2}-12ab+9b^{2}$$

Combine 12a^{2} and 4a^{2} to get 16a^{2}.

$$16a^{2}+12ab-9b^{2}-12ab+9b^{2}$$

Combine 12ab and -12ab to get 0.

$$16a^{2}-9b^{2}+9b^{2}$$

Combine -9b^{2} and 9b^{2} to get 0.

$$16a^{2}$$