Solve 1/3x(x+1)(x+2)+(x+1)(x+2)

$$\frac{ 1 }{ 3 } x(x+1)(x+2)+(x+1)(x+2)$$

Evaluate

$$\frac{\left(x+1\right)\left(x+2\right)\left(x+3\right)}{3}$$

Solution Steps

Use the distributive property to multiply \frac{1}{3}x by x+1.

$$\left(\frac{1}{3}xx+\frac{1}{3}x\right)\left(x+2\right)+\left(x+1\right)\left(x+2\right)$$

Multiply x and x to get x^{2}.

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

Apply the distributive property by multiplying each term of \frac{1}{3}x^{2}+\frac{1}{3}x by each term of x+2.

$$\frac{1}{3}x^{2}x+\frac{1}{3}x^{2}\times 2+\frac{1}{3}xx+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{3}x^{3}+\frac{1}{3}x^{2}\times 2+\frac{1}{3}xx+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Multiply x and x to get x^{2}.

$$\frac{1}{3}x^{3}+\frac{1}{3}x^{2}\times 2+\frac{1}{3}x^{2}+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$\frac{1}{3}x^{3}+\frac{2}{3}x^{2}+\frac{1}{3}x^{2}+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Combine \frac{2}{3}x^{2} and \frac{1}{3}x^{2} to get x^{2}.

$$\frac{1}{3}x^{3}+x^{2}+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

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

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

$$\frac{1}{3}x^{3}+x^{2}+\frac{2}{3}x+x^{2}+2x+x+2$$

Combine 2x and x to get 3x.

$$\frac{1}{3}x^{3}+x^{2}+\frac{2}{3}x+x^{2}+3x+2$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$\frac{1}{3}x^{3}+2x^{2}+\frac{2}{3}x+3x+2$$

Combine \frac{2}{3}x and 3x to get \frac{11}{3}x.

$$\frac{1}{3}x^{3}+2x^{2}+\frac{11}{3}x+2$$

Expand

$$\frac{x^{3}}{3}+2x^{2}+\frac{11x}{3}+2$$

Solution Steps

Use the distributive property to multiply \frac{1}{3}x by x+1.

$$\left(\frac{1}{3}xx+\frac{1}{3}x\right)\left(x+2\right)+\left(x+1\right)\left(x+2\right)$$

Multiply x and x to get x^{2}.

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

Apply the distributive property by multiplying each term of \frac{1}{3}x^{2}+\frac{1}{3}x by each term of x+2.

$$\frac{1}{3}x^{2}x+\frac{1}{3}x^{2}\times 2+\frac{1}{3}xx+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{3}x^{3}+\frac{1}{3}x^{2}\times 2+\frac{1}{3}xx+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Multiply x and x to get x^{2}.

$$\frac{1}{3}x^{3}+\frac{1}{3}x^{2}\times 2+\frac{1}{3}x^{2}+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$\frac{1}{3}x^{3}+\frac{2}{3}x^{2}+\frac{1}{3}x^{2}+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Combine \frac{2}{3}x^{2} and \frac{1}{3}x^{2} to get x^{2}.

$$\frac{1}{3}x^{3}+x^{2}+\frac{1}{3}x\times 2+\left(x+1\right)\left(x+2\right)$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

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

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

$$\frac{1}{3}x^{3}+x^{2}+\frac{2}{3}x+x^{2}+2x+x+2$$

Combine 2x and x to get 3x.

$$\frac{1}{3}x^{3}+x^{2}+\frac{2}{3}x+x^{2}+3x+2$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$\frac{1}{3}x^{3}+2x^{2}+\frac{2}{3}x+3x+2$$

Combine \frac{2}{3}x and 3x to get \frac{11}{3}x.

$$\frac{1}{3}x^{3}+2x^{2}+\frac{11}{3}x+2$$