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

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

Evaluate

$$3b^{2}-8ab-a^{2}$$

Solution Steps

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

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

Combine ab and -6ba to get -5ab.

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

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

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

Combine 2ab and ba to get 3ab.

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

To find the opposite of 3ab-2a^{2}-b^{2}, find the opposite of each term.

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

The opposite of -2a^{2} is 2a^{2}.

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

The opposite of -b^{2} is b^{2}.

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

Combine -5ab and -3ab to get -8ab.

$$-8ab-3a^{2}+2b^{2}+2a^{2}+b^{2}$$

Combine -3a^{2} and 2a^{2} to get -a^{2}.

$$-8ab-a^{2}+2b^{2}+b^{2}$$

Combine 2b^{2} and b^{2} to get 3b^{2}.

$$-8ab-a^{2}+3b^{2}$$

Expand

$$3b^{2}-8ab-a^{2}$$

Solution Steps

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

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

Combine ab and -6ba to get -5ab.

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

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

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

Combine 2ab and ba to get 3ab.

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

To find the opposite of 3ab-2a^{2}-b^{2}, find the opposite of each term.

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

The opposite of -2a^{2} is 2a^{2}.

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

The opposite of -b^{2} is b^{2}.

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

Combine -5ab and -3ab to get -8ab.

$$-8ab-3a^{2}+2b^{2}+2a^{2}+b^{2}$$

Combine -3a^{2} and 2a^{2} to get -a^{2}.

$$-8ab-a^{2}+2b^{2}+b^{2}$$

Combine 2b^{2} and b^{2} to get 3b^{2}.

$$-8ab-a^{2}+3b^{2}$$