Solve (a+b)^2+6(a+b)(2a+b)+9(2a+b)^2

$$( a + b ) ^ { 2 } + 6 ( a + b ) ( 2 a + b ) + 9 ( 2 a + b ) ^ { 2 }$$

Evaluate

$$\left(7a+4b\right)^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 6 by a+b.

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

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

$$a^{2}+2ab+b^{2}+12a^{2}+18ab+6b^{2}+9\left(2a+b\right)^{2}$$

Combine a^{2} and 12a^{2} to get 13a^{2}.

$$13a^{2}+2ab+b^{2}+18ab+6b^{2}+9\left(2a+b\right)^{2}$$

Combine 2ab and 18ab to get 20ab.

$$13a^{2}+20ab+b^{2}+6b^{2}+9\left(2a+b\right)^{2}$$

Combine b^{2} and 6b^{2} to get 7b^{2}.

$$13a^{2}+20ab+7b^{2}+9\left(2a+b\right)^{2}$$

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

$$13a^{2}+20ab+7b^{2}+9\left(4a^{2}+4ab+b^{2}\right)$$

Use the distributive property to multiply 9 by 4a^{2}+4ab+b^{2}.

$$13a^{2}+20ab+7b^{2}+36a^{2}+36ab+9b^{2}$$

Combine 13a^{2} and 36a^{2} to get 49a^{2}.

$$49a^{2}+20ab+7b^{2}+36ab+9b^{2}$$

Combine 20ab and 36ab to get 56ab.

$$49a^{2}+56ab+7b^{2}+9b^{2}$$

Combine 7b^{2} and 9b^{2} to get 16b^{2}.

$$49a^{2}+56ab+16b^{2}$$

Expand

$$49a^{2}+56ab+16b^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 6 by a+b.

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

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

$$a^{2}+2ab+b^{2}+12a^{2}+18ab+6b^{2}+9\left(2a+b\right)^{2}$$

Combine a^{2} and 12a^{2} to get 13a^{2}.

$$13a^{2}+2ab+b^{2}+18ab+6b^{2}+9\left(2a+b\right)^{2}$$

Combine 2ab and 18ab to get 20ab.

$$13a^{2}+20ab+b^{2}+6b^{2}+9\left(2a+b\right)^{2}$$

Combine b^{2} and 6b^{2} to get 7b^{2}.

$$13a^{2}+20ab+7b^{2}+9\left(2a+b\right)^{2}$$

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

$$13a^{2}+20ab+7b^{2}+9\left(4a^{2}+4ab+b^{2}\right)$$

Use the distributive property to multiply 9 by 4a^{2}+4ab+b^{2}.

$$13a^{2}+20ab+7b^{2}+36a^{2}+36ab+9b^{2}$$

Combine 13a^{2} and 36a^{2} to get 49a^{2}.

$$49a^{2}+20ab+7b^{2}+36ab+9b^{2}$$

Combine 20ab and 36ab to get 56ab.

$$49a^{2}+56ab+7b^{2}+9b^{2}$$

Combine 7b^{2} and 9b^{2} to get 16b^{2}.

$$49a^{2}+56ab+16b^{2}$$