Solve (a-2)(a-3)-(a+1)(a+2)(a+3)]:(-4)

$$( a - 2 ) ( a - 3 ) - ( a + 1 ) ( a + 2 ) ( a + 3 ) ] : ( - 4 )$$

Evaluate

$$-a\left(a^{2}+5a+16\right)$$

Solution Steps

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

$$a^{2}-3a-2a+6-\left(a+1\right)\left(a+2\right)\left(a+3\right)$$

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

$$a^{2}-5a+6-\left(a+1\right)\left(a+2\right)\left(a+3\right)$$

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

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

Combine 2a and a to get 3a.

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

Apply the distributive property by multiplying each term of a^{2}+3a+2 by each term of a+3.

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

Combine 3a^{2} and 3a^{2} to get 6a^{2}.

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

Combine 9a and 2a to get 11a.

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

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

$$a^{2}-5a+6-a^{3}-6a^{2}-11a-6$$

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

$$-5a^{2}-5a+6-a^{3}-11a-6$$

Combine -5a and -11a to get -16a.

$$-5a^{2}-16a+6-a^{3}-6$$

Subtract 6 from 6 to get 0.

$$-5a^{2}-16a-a^{3}$$

Expand

$$-a^{3}-5a^{2}-16a$$

Solution Steps

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

$$a^{2}-3a-2a+6-\left(a+1\right)\left(a+2\right)\left(a+3\right)$$

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

$$a^{2}-5a+6-\left(a+1\right)\left(a+2\right)\left(a+3\right)$$

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

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

Combine 2a and a to get 3a.

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

Apply the distributive property by multiplying each term of a^{2}+3a+2 by each term of a+3.

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

Combine 3a^{2} and 3a^{2} to get 6a^{2}.

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

Combine 9a and 2a to get 11a.

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

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

$$a^{2}-5a+6-a^{3}-6a^{2}-11a-6$$

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

$$-5a^{2}-5a+6-a^{3}-11a-6$$

Combine -5a and -11a to get -16a.

$$-5a^{2}-16a+6-a^{3}-6$$

Subtract 6 from 6 to get 0.

$$-5a^{2}-16a-a^{3}$$