Solve a-[(2a+1)(a-2)+(a-3)(a+1)-(3a-2)(a-3)]

$$a - [ ( 2 a + 1 ) ( a - 2 ) + ( a - 3 ) ( a + 1 ) - ( 3 a - 2 ) ( a - 3 ) ]$$

Evaluate

$$11-5a$$

Solution Steps

Apply the distributive property by multiplying each term of 2a+1 by each term of a-2.

$$a-\left(2a^{2}-4a+a-2+\left(a-3\right)\left(a+1\right)-\left(3a-2\right)\left(a-3\right)\right)$$

Combine -4a and a to get -3a.

$$a-\left(2a^{2}-3a-2+\left(a-3\right)\left(a+1\right)-\left(3a-2\right)\left(a-3\right)\right)$$

Apply the distributive property by multiplying each term of a-3 by each term of a+1.

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

Combine a and -3a to get -2a.

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

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

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

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

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

Subtract 3 from -2 to get -5.

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

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

$$a-\left(3a^{2}-5a-5-\left(3a^{2}-9a-2a+6\right)\right)$$

Combine -9a and -2a to get -11a.

$$a-\left(3a^{2}-5a-5-\left(3a^{2}-11a+6\right)\right)$$

To find the opposite of 3a^{2}-11a+6, find the opposite of each term.

$$a-\left(3a^{2}-5a-5-3a^{2}-\left(-11a\right)-6\right)$$

The opposite of -11a is 11a.

$$a-\left(3a^{2}-5a-5-3a^{2}+11a-6\right)$$

Combine 3a^{2} and -3a^{2} to get 0.

$$a-\left(-5a-5+11a-6\right)$$

Combine -5a and 11a to get 6a.

$$a-\left(6a-5-6\right)$$

Subtract 6 from -5 to get -11.

$$a-\left(6a-11\right)$$

To find the opposite of 6a-11, find the opposite of each term.

$$a-6a-\left(-11\right)$$

The opposite of -11 is 11.

$$a-6a+11$$

Combine a and -6a to get -5a.

$$-5a+11$$

Expand

$$11-5a$$

Solution Steps

Apply the distributive property by multiplying each term of 2a+1 by each term of a-2.

$$a-\left(2a^{2}-4a+a-2+\left(a-3\right)\left(a+1\right)-\left(3a-2\right)\left(a-3\right)\right)$$

Combine -4a and a to get -3a.

$$a-\left(2a^{2}-3a-2+\left(a-3\right)\left(a+1\right)-\left(3a-2\right)\left(a-3\right)\right)$$

Apply the distributive property by multiplying each term of a-3 by each term of a+1.

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

Combine a and -3a to get -2a.

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

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

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

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

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

Subtract 3 from -2 to get -5.

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

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

$$a-\left(3a^{2}-5a-5-\left(3a^{2}-9a-2a+6\right)\right)$$

Combine -9a and -2a to get -11a.

$$a-\left(3a^{2}-5a-5-\left(3a^{2}-11a+6\right)\right)$$

To find the opposite of 3a^{2}-11a+6, find the opposite of each term.

$$a-\left(3a^{2}-5a-5-3a^{2}-\left(-11a\right)-6\right)$$

The opposite of -11a is 11a.

$$a-\left(3a^{2}-5a-5-3a^{2}+11a-6\right)$$

Combine 3a^{2} and -3a^{2} to get 0.

$$a-\left(-5a-5+11a-6\right)$$

Combine -5a and 11a to get 6a.

$$a-\left(6a-5-6\right)$$

Subtract 6 from -5 to get -11.

$$a-\left(6a-11\right)$$

To find the opposite of 6a-11, find the opposite of each term.

$$a-6a-\left(-11\right)$$

The opposite of -11 is 11.

$$a-6a+11$$

Combine a and -6a to get -5a.

$$-5a+11$$