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

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

Evaluate

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

Solution Steps

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

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

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

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

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

$$\frac{a^{3}-3a^{2}-3a^{2}+9a+2a-6-\left(a+1\right)\left(a+2\right)\left(a+3\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+9a+2a-6-\left(a+1\right)\left(a+2\right)\left(a+3\right)}{-4}$$

Combine 9a and 2a to get 11a.

$$\frac{a^{3}-6a^{2}+11a-6-\left(a+1\right)\left(a+2\right)\left(a+3\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{2}+2a+a+2\right)\left(a+3\right)}{-4}$$

Combine 2a and a to get 3a.

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{2}+3a+2\right)\left(a+3\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{3}+3a^{2}+3a^{2}+9a+2a+6\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{3}+6a^{2}+9a+2a+6\right)}{-4}$$

Combine 9a and 2a to get 11a.

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{3}+6a^{2}+11a+6\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-a^{3}-6a^{2}-11a-6}{-4}$$

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

$$\frac{-6a^{2}+11a-6-6a^{2}-11a-6}{-4}$$

Combine -6a^{2} and -6a^{2} to get -12a^{2}.

$$\frac{-12a^{2}+11a-6-11a-6}{-4}$$

Combine 11a and -11a to get 0.

$$\frac{-12a^{2}-6-6}{-4}$$

Subtract 6 from -6 to get -12.

$$\frac{-12a^{2}-12}{-4}$$

Expand

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

Solution Steps

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

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

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

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

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

$$\frac{a^{3}-3a^{2}-3a^{2}+9a+2a-6-\left(a+1\right)\left(a+2\right)\left(a+3\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+9a+2a-6-\left(a+1\right)\left(a+2\right)\left(a+3\right)}{-4}$$

Combine 9a and 2a to get 11a.

$$\frac{a^{3}-6a^{2}+11a-6-\left(a+1\right)\left(a+2\right)\left(a+3\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{2}+2a+a+2\right)\left(a+3\right)}{-4}$$

Combine 2a and a to get 3a.

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{2}+3a+2\right)\left(a+3\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{3}+3a^{2}+3a^{2}+9a+2a+6\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{3}+6a^{2}+9a+2a+6\right)}{-4}$$

Combine 9a and 2a to get 11a.

$$\frac{a^{3}-6a^{2}+11a-6-\left(a^{3}+6a^{2}+11a+6\right)}{-4}$$

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

$$\frac{a^{3}-6a^{2}+11a-6-a^{3}-6a^{2}-11a-6}{-4}$$

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

$$\frac{-6a^{2}+11a-6-6a^{2}-11a-6}{-4}$$

Combine -6a^{2} and -6a^{2} to get -12a^{2}.

$$\frac{-12a^{2}+11a-6-11a-6}{-4}$$

Combine 11a and -11a to get 0.

$$\frac{-12a^{2}-6-6}{-4}$$

Subtract 6 from -6 to get -12.

$$\frac{-12a^{2}-12}{-4}$$