Solve left(x+1right)left(x+2right)left(x+3right)(x+4)(x+5)(x+6)

$$\left( x+1 \right) \left( x+2 \right) \left( x+3 \right) (x+4)(x+5)(x+6)$$

Expand

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1624x^{2}+1764x+720$$

Solution Steps

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

$$\left(x^{2}+2x+x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Combine 2x and x to get 3x.

$$\left(x^{2}+3x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

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

$$\left(x^{3}+3x^{2}+3x^{2}+9x+2x+6\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

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

$$\left(x^{3}+6x^{2}+9x+2x+6\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Combine 9x and 2x to get 11x.

$$\left(x^{3}+6x^{2}+11x+6\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Apply the distributive property by multiplying each term of x^{3}+6x^{2}+11x+6 by each term of x+4.

$$\left(x^{4}+4x^{3}+6x^{3}+24x^{2}+11x^{2}+44x+6x+24\right)\left(x+5\right)\left(x+6\right)$$

Combine 4x^{3} and 6x^{3} to get 10x^{3}.

$$\left(x^{4}+10x^{3}+24x^{2}+11x^{2}+44x+6x+24\right)\left(x+5\right)\left(x+6\right)$$

Combine 24x^{2} and 11x^{2} to get 35x^{2}.

$$\left(x^{4}+10x^{3}+35x^{2}+44x+6x+24\right)\left(x+5\right)\left(x+6\right)$$

Combine 44x and 6x to get 50x.

$$\left(x^{4}+10x^{3}+35x^{2}+50x+24\right)\left(x+5\right)\left(x+6\right)$$

Apply the distributive property by multiplying each term of x^{4}+10x^{3}+35x^{2}+50x+24 by each term of x+5.

$$\left(x^{5}+5x^{4}+10x^{4}+50x^{3}+35x^{3}+175x^{2}+50x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 5x^{4} and 10x^{4} to get 15x^{4}.

$$\left(x^{5}+15x^{4}+50x^{3}+35x^{3}+175x^{2}+50x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 50x^{3} and 35x^{3} to get 85x^{3}.

$$\left(x^{5}+15x^{4}+85x^{3}+175x^{2}+50x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 175x^{2} and 50x^{2} to get 225x^{2}.

$$\left(x^{5}+15x^{4}+85x^{3}+225x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 250x and 24x to get 274x.

$$\left(x^{5}+15x^{4}+85x^{3}+225x^{2}+274x+120\right)\left(x+6\right)$$

Apply the distributive property by multiplying each term of x^{5}+15x^{4}+85x^{3}+225x^{2}+274x+120 by each term of x+6.

$$x^{6}+6x^{5}+15x^{5}+90x^{4}+85x^{4}+510x^{3}+225x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 6x^{5} and 15x^{5} to get 21x^{5}.

$$x^{6}+21x^{5}+90x^{4}+85x^{4}+510x^{3}+225x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 90x^{4} and 85x^{4} to get 175x^{4}.

$$x^{6}+21x^{5}+175x^{4}+510x^{3}+225x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 510x^{3} and 225x^{3} to get 735x^{3}.

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 1350x^{2} and 274x^{2} to get 1624x^{2}.

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1624x^{2}+1644x+120x+720$$

Combine 1644x and 120x to get 1764x.

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1624x^{2}+1764x+720$$

Evaluate

$$\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Solution Steps

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

$$\left(x^{2}+2x+x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Combine 2x and x to get 3x.

$$\left(x^{2}+3x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

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

$$\left(x^{3}+3x^{2}+3x^{2}+9x+2x+6\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

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

$$\left(x^{3}+6x^{2}+9x+2x+6\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Combine 9x and 2x to get 11x.

$$\left(x^{3}+6x^{2}+11x+6\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Apply the distributive property by multiplying each term of x^{3}+6x^{2}+11x+6 by each term of x+4.

$$\left(x^{4}+4x^{3}+6x^{3}+24x^{2}+11x^{2}+44x+6x+24\right)\left(x+5\right)\left(x+6\right)$$

Combine 4x^{3} and 6x^{3} to get 10x^{3}.

$$\left(x^{4}+10x^{3}+24x^{2}+11x^{2}+44x+6x+24\right)\left(x+5\right)\left(x+6\right)$$

Combine 24x^{2} and 11x^{2} to get 35x^{2}.

$$\left(x^{4}+10x^{3}+35x^{2}+44x+6x+24\right)\left(x+5\right)\left(x+6\right)$$

Combine 44x and 6x to get 50x.

$$\left(x^{4}+10x^{3}+35x^{2}+50x+24\right)\left(x+5\right)\left(x+6\right)$$

Apply the distributive property by multiplying each term of x^{4}+10x^{3}+35x^{2}+50x+24 by each term of x+5.

$$\left(x^{5}+5x^{4}+10x^{4}+50x^{3}+35x^{3}+175x^{2}+50x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 5x^{4} and 10x^{4} to get 15x^{4}.

$$\left(x^{5}+15x^{4}+50x^{3}+35x^{3}+175x^{2}+50x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 50x^{3} and 35x^{3} to get 85x^{3}.

$$\left(x^{5}+15x^{4}+85x^{3}+175x^{2}+50x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 175x^{2} and 50x^{2} to get 225x^{2}.

$$\left(x^{5}+15x^{4}+85x^{3}+225x^{2}+250x+24x+120\right)\left(x+6\right)$$

Combine 250x and 24x to get 274x.

$$\left(x^{5}+15x^{4}+85x^{3}+225x^{2}+274x+120\right)\left(x+6\right)$$

Apply the distributive property by multiplying each term of x^{5}+15x^{4}+85x^{3}+225x^{2}+274x+120 by each term of x+6.

$$x^{6}+6x^{5}+15x^{5}+90x^{4}+85x^{4}+510x^{3}+225x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 6x^{5} and 15x^{5} to get 21x^{5}.

$$x^{6}+21x^{5}+90x^{4}+85x^{4}+510x^{3}+225x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 90x^{4} and 85x^{4} to get 175x^{4}.

$$x^{6}+21x^{5}+175x^{4}+510x^{3}+225x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 510x^{3} and 225x^{3} to get 735x^{3}.

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1350x^{2}+274x^{2}+1644x+120x+720$$

Combine 1350x^{2} and 274x^{2} to get 1624x^{2}.

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1624x^{2}+1644x+120x+720$$

Combine 1644x and 120x to get 1764x.

$$x^{6}+21x^{5}+175x^{4}+735x^{3}+1624x^{2}+1764x+720$$