Solve (2a+b)^2+5a(a+b)-(3a-b)^2

$$( 2 a + b ) ^ { 2 } + 5 a ( a + b ) - ( 3 a - b ) ^ { 2 }$$

Evaluate

$$15ab$$

Solution Steps

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

$$4a^{2}+4ab+b^{2}+5a\left(a+b\right)-\left(3a-b\right)^{2}$$

Use the distributive property to multiply 5a by a+b.

$$4a^{2}+4ab+b^{2}+5a^{2}+5ab-\left(3a-b\right)^{2}$$

Combine 4a^{2} and 5a^{2} to get 9a^{2}.

$$9a^{2}+4ab+b^{2}+5ab-\left(3a-b\right)^{2}$$

Combine 4ab and 5ab to get 9ab.

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

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

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

To find the opposite of 9a^{2}-6ab+b^{2}, find the opposite of each term.

$$9a^{2}+9ab+b^{2}-9a^{2}+6ab-b^{2}$$

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

$$9ab+b^{2}+6ab-b^{2}$$

Combine 9ab and 6ab to get 15ab.

$$15ab+b^{2}-b^{2}$$

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

$$15ab$$

Expand

$$15ab$$

Solution Steps

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

$$4a^{2}+4ab+b^{2}+5a\left(a+b\right)-\left(3a-b\right)^{2}$$

Use the distributive property to multiply 5a by a+b.

$$4a^{2}+4ab+b^{2}+5a^{2}+5ab-\left(3a-b\right)^{2}$$

Combine 4a^{2} and 5a^{2} to get 9a^{2}.

$$9a^{2}+4ab+b^{2}+5ab-\left(3a-b\right)^{2}$$

Combine 4ab and 5ab to get 9ab.

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

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

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

To find the opposite of 9a^{2}-6ab+b^{2}, find the opposite of each term.

$$9a^{2}+9ab+b^{2}-9a^{2}+6ab-b^{2}$$

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

$$9ab+b^{2}+6ab-b^{2}$$

Combine 9ab and 6ab to get 15ab.

$$15ab+b^{2}-b^{2}$$

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

$$15ab$$