Solve a(a+1)(a+2)(a+3)+1

$$a ( a + 1 ) ( a + 2 ) ( a + 3 ) + 1$$

Expand

$$a^{4}+6a^{3}+11a^{2}+6a+1$$

Solution Steps

Use the distributive property to multiply a by a+1.

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

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

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

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

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

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

$$a^{4}+3a^{3}+3a^{3}+9a^{2}+2a^{2}+6a+1$$

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

$$a^{4}+6a^{3}+9a^{2}+2a^{2}+6a+1$$

Combine 9a^{2} and 2a^{2} to get 11a^{2}.

$$a^{4}+6a^{3}+11a^{2}+6a+1$$

Evaluate

$$a\left(a+1\right)\left(a+2\right)\left(a+3\right)+1$$

Solution Steps

Use the distributive property to multiply a by a+1.

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

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

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

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

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

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

$$a^{4}+3a^{3}+3a^{3}+9a^{2}+2a^{2}+6a+1$$

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

$$a^{4}+6a^{3}+9a^{2}+2a^{2}+6a+1$$

Combine 9a^{2} and 2a^{2} to get 11a^{2}.

$$a^{4}+6a^{3}+11a^{2}+6a+1$$