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

$$( 3 a + b ) ( 2 a + b ) - ( a + b ) ( a + b )$$

Evaluate

$$a\left(5a+3b\right)$$

Solution Steps

Multiply a+b and a+b to get \left(a+b\right)^{2}.

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

Apply the distributive property by multiplying each term of 3a+b by each term of 2a+b.

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

Combine 3ab and 2ba to get 5ab.

$$6a^{2}+5ab+b^{2}-\left(a+b\right)^{2}$$

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

$$6a^{2}+5ab+b^{2}-\left(a^{2}+2ab+b^{2}\right)$$

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

$$6a^{2}+5ab+b^{2}-a^{2}-2ab-b^{2}$$

Combine 6a^{2} and -a^{2} to get 5a^{2}.

$$5a^{2}+5ab+b^{2}-2ab-b^{2}$$

Combine 5ab and -2ab to get 3ab.

$$5a^{2}+3ab+b^{2}-b^{2}$$

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

$$5a^{2}+3ab$$

Expand

$$5a^{2}+3ab$$

Solution Steps

Multiply a+b and a+b to get \left(a+b\right)^{2}.

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

Apply the distributive property by multiplying each term of 3a+b by each term of 2a+b.

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

Combine 3ab and 2ba to get 5ab.

$$6a^{2}+5ab+b^{2}-\left(a+b\right)^{2}$$

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

$$6a^{2}+5ab+b^{2}-\left(a^{2}+2ab+b^{2}\right)$$

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

$$6a^{2}+5ab+b^{2}-a^{2}-2ab-b^{2}$$

Combine 6a^{2} and -a^{2} to get 5a^{2}.

$$5a^{2}+5ab+b^{2}-2ab-b^{2}$$

Combine 5ab and -2ab to get 3ab.

$$5a^{2}+3ab+b^{2}-b^{2}$$

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

$$5a^{2}+3ab$$