Solve (x+1)(x+3)(x+4)-x(x+1)(x+2)

$$( x + 1 ) ( x + 3 ) ( x + 4 ) - x ( x + 1 ) ( x + 2 )$$

Evaluate

$$\left(x+1\right)\left(5x+12\right)$$

Solution Steps

Apply the distributive property by multiplying each term of x+1 by each term of x+3.

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

Combine 3x and x to get 4x.

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

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

$$x^{3}+4x^{2}+4x^{2}+16x+3x+12-x\left(x+1\right)\left(x+2\right)$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$x^{3}+8x^{2}+16x+3x+12-x\left(x+1\right)\left(x+2\right)$$

Combine 16x and 3x to get 19x.

$$x^{3}+8x^{2}+19x+12-x\left(x+1\right)\left(x+2\right)$$

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

$$x^{3}+8x^{2}+19x+12-\left(x^{2}+x\right)\left(x+2\right)$$

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

$$x^{3}+8x^{2}+19x+12-\left(x^{3}+2x^{2}+x^{2}+2x\right)$$

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

$$x^{3}+8x^{2}+19x+12-\left(x^{3}+3x^{2}+2x\right)$$

To find the opposite of x^{3}+3x^{2}+2x, find the opposite of each term.

$$x^{3}+8x^{2}+19x+12-x^{3}-3x^{2}-2x$$

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

$$8x^{2}+19x+12-3x^{2}-2x$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}+19x+12-2x$$

Combine 19x and -2x to get 17x.

$$5x^{2}+17x+12$$

Expand

$$5x^{2}+17x+12$$

Solution Steps

Apply the distributive property by multiplying each term of x+1 by each term of x+3.

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

Combine 3x and x to get 4x.

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

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

$$x^{3}+4x^{2}+4x^{2}+16x+3x+12-x\left(x+1\right)\left(x+2\right)$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$x^{3}+8x^{2}+16x+3x+12-x\left(x+1\right)\left(x+2\right)$$

Combine 16x and 3x to get 19x.

$$x^{3}+8x^{2}+19x+12-x\left(x+1\right)\left(x+2\right)$$

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

$$x^{3}+8x^{2}+19x+12-\left(x^{2}+x\right)\left(x+2\right)$$

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

$$x^{3}+8x^{2}+19x+12-\left(x^{3}+2x^{2}+x^{2}+2x\right)$$

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

$$x^{3}+8x^{2}+19x+12-\left(x^{3}+3x^{2}+2x\right)$$

To find the opposite of x^{3}+3x^{2}+2x, find the opposite of each term.

$$x^{3}+8x^{2}+19x+12-x^{3}-3x^{2}-2x$$

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

$$8x^{2}+19x+12-3x^{2}-2x$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}+19x+12-2x$$

Combine 19x and -2x to get 17x.

$$5x^{2}+17x+12$$