Solve {l}{(x-2)(x-3)(x-4)}{+2(x-1)(x-3)(x-4)}{+3(x-1)(x-2)(x-4)}{+4(x-1)(x-2)(x-3)}

$$\left. \begin{array} { l } { ( x - 2 ) ( x - 3 ) ( x - 4 ) } \\ { + 2 ( x - 1 ) ( x - 3 ) ( x - 4 ) } \\ { + 3 ( x - 1 ) ( x - 2 ) ( x - 4 ) } \\ { + 4 ( x - 1 ) ( x - 2 ) ( x - 3 ) } \end{array} \right.$$

Evaluate

$$10x^{3}-70x^{2}+150x-96$$

Solution Steps

Apply the distributive property by multiplying each term of x-2 by each term of x-3.

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

Combine -3x and -2x to get -5x.

$$\left(x^{2}-5x+6\right)\left(x-4\right)+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x^{2}-5x+6 by each term of x-4.

$$x^{3}-4x^{2}-5x^{2}+20x+6x-24+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -4x^{2} and -5x^{2} to get -9x^{2}.

$$x^{3}-9x^{2}+20x+6x-24+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 20x and 6x to get 26x.

$$x^{3}-9x^{2}+26x-24+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 2 by x-1.

$$x^{3}-9x^{2}+26x-24+\left(2x-2\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 2x-2 by each term of x-3.

$$x^{3}-9x^{2}+26x-24+\left(2x^{2}-6x-2x+6\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -6x and -2x to get -8x.

$$x^{3}-9x^{2}+26x-24+\left(2x^{2}-8x+6\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 2x^{2}-8x+6 by each term of x-4.

$$x^{3}-9x^{2}+26x-24+2x^{3}-8x^{2}-8x^{2}+32x+6x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -8x^{2} and -8x^{2} to get -16x^{2}.

$$x^{3}-9x^{2}+26x-24+2x^{3}-16x^{2}+32x+6x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 32x and 6x to get 38x.

$$x^{3}-9x^{2}+26x-24+2x^{3}-16x^{2}+38x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

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

$$3x^{3}-9x^{2}+26x-24-16x^{2}+38x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -9x^{2} and -16x^{2} to get -25x^{2}.

$$3x^{3}-25x^{2}+26x-24+38x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 26x and 38x to get 64x.

$$3x^{3}-25x^{2}+64x-24-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Subtract 24 from -24 to get -48.

$$3x^{3}-25x^{2}+64x-48+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 3 by x-1.

$$3x^{3}-25x^{2}+64x-48+\left(3x-3\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 3x-3 by each term of x-2.

$$3x^{3}-25x^{2}+64x-48+\left(3x^{2}-6x-3x+6\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -6x and -3x to get -9x.

$$3x^{3}-25x^{2}+64x-48+\left(3x^{2}-9x+6\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 3x^{2}-9x+6 by each term of x-4.

$$3x^{3}-25x^{2}+64x-48+3x^{3}-12x^{2}-9x^{2}+36x+6x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -12x^{2} and -9x^{2} to get -21x^{2}.

$$3x^{3}-25x^{2}+64x-48+3x^{3}-21x^{2}+36x+6x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 36x and 6x to get 42x.

$$3x^{3}-25x^{2}+64x-48+3x^{3}-21x^{2}+42x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

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

$$6x^{3}-25x^{2}+64x-48-21x^{2}+42x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -25x^{2} and -21x^{2} to get -46x^{2}.

$$6x^{3}-46x^{2}+64x-48+42x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 64x and 42x to get 106x.

$$6x^{3}-46x^{2}+106x-48-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Subtract 24 from -48 to get -72.

$$6x^{3}-46x^{2}+106x-72+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 4 by x-1.

$$6x^{3}-46x^{2}+106x-72+\left(4x-4\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 4x-4 by each term of x-2.

$$6x^{3}-46x^{2}+106x-72+\left(4x^{2}-8x-4x+8\right)\left(x-3\right)$$

Combine -8x and -4x to get -12x.

$$6x^{3}-46x^{2}+106x-72+\left(4x^{2}-12x+8\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 4x^{2}-12x+8 by each term of x-3.

$$6x^{3}-46x^{2}+106x-72+4x^{3}-12x^{2}-12x^{2}+36x+8x-24$$

Combine -12x^{2} and -12x^{2} to get -24x^{2}.

$$6x^{3}-46x^{2}+106x-72+4x^{3}-24x^{2}+36x+8x-24$$

Combine 36x and 8x to get 44x.

$$6x^{3}-46x^{2}+106x-72+4x^{3}-24x^{2}+44x-24$$

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

$$10x^{3}-46x^{2}+106x-72-24x^{2}+44x-24$$

Combine -46x^{2} and -24x^{2} to get -70x^{2}.

$$10x^{3}-70x^{2}+106x-72+44x-24$$

Combine 106x and 44x to get 150x.

$$10x^{3}-70x^{2}+150x-72-24$$

Subtract 24 from -72 to get -96.

$$10x^{3}-70x^{2}+150x-96$$

Expand

$$10x^{3}-70x^{2}+150x-96$$

Solution Steps

Apply the distributive property by multiplying each term of x-2 by each term of x-3.

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

Combine -3x and -2x to get -5x.

$$\left(x^{2}-5x+6\right)\left(x-4\right)+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x^{2}-5x+6 by each term of x-4.

$$x^{3}-4x^{2}-5x^{2}+20x+6x-24+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -4x^{2} and -5x^{2} to get -9x^{2}.

$$x^{3}-9x^{2}+20x+6x-24+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 20x and 6x to get 26x.

$$x^{3}-9x^{2}+26x-24+2\left(x-1\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 2 by x-1.

$$x^{3}-9x^{2}+26x-24+\left(2x-2\right)\left(x-3\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 2x-2 by each term of x-3.

$$x^{3}-9x^{2}+26x-24+\left(2x^{2}-6x-2x+6\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -6x and -2x to get -8x.

$$x^{3}-9x^{2}+26x-24+\left(2x^{2}-8x+6\right)\left(x-4\right)+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 2x^{2}-8x+6 by each term of x-4.

$$x^{3}-9x^{2}+26x-24+2x^{3}-8x^{2}-8x^{2}+32x+6x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -8x^{2} and -8x^{2} to get -16x^{2}.

$$x^{3}-9x^{2}+26x-24+2x^{3}-16x^{2}+32x+6x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 32x and 6x to get 38x.

$$x^{3}-9x^{2}+26x-24+2x^{3}-16x^{2}+38x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

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

$$3x^{3}-9x^{2}+26x-24-16x^{2}+38x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -9x^{2} and -16x^{2} to get -25x^{2}.

$$3x^{3}-25x^{2}+26x-24+38x-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 26x and 38x to get 64x.

$$3x^{3}-25x^{2}+64x-24-24+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Subtract 24 from -24 to get -48.

$$3x^{3}-25x^{2}+64x-48+3\left(x-1\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 3 by x-1.

$$3x^{3}-25x^{2}+64x-48+\left(3x-3\right)\left(x-2\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 3x-3 by each term of x-2.

$$3x^{3}-25x^{2}+64x-48+\left(3x^{2}-6x-3x+6\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -6x and -3x to get -9x.

$$3x^{3}-25x^{2}+64x-48+\left(3x^{2}-9x+6\right)\left(x-4\right)+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 3x^{2}-9x+6 by each term of x-4.

$$3x^{3}-25x^{2}+64x-48+3x^{3}-12x^{2}-9x^{2}+36x+6x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -12x^{2} and -9x^{2} to get -21x^{2}.

$$3x^{3}-25x^{2}+64x-48+3x^{3}-21x^{2}+36x+6x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 36x and 6x to get 42x.

$$3x^{3}-25x^{2}+64x-48+3x^{3}-21x^{2}+42x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

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

$$6x^{3}-25x^{2}+64x-48-21x^{2}+42x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -25x^{2} and -21x^{2} to get -46x^{2}.

$$6x^{3}-46x^{2}+64x-48+42x-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 64x and 42x to get 106x.

$$6x^{3}-46x^{2}+106x-48-24+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Subtract 24 from -48 to get -72.

$$6x^{3}-46x^{2}+106x-72+4\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 4 by x-1.

$$6x^{3}-46x^{2}+106x-72+\left(4x-4\right)\left(x-2\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 4x-4 by each term of x-2.

$$6x^{3}-46x^{2}+106x-72+\left(4x^{2}-8x-4x+8\right)\left(x-3\right)$$

Combine -8x and -4x to get -12x.

$$6x^{3}-46x^{2}+106x-72+\left(4x^{2}-12x+8\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of 4x^{2}-12x+8 by each term of x-3.

$$6x^{3}-46x^{2}+106x-72+4x^{3}-12x^{2}-12x^{2}+36x+8x-24$$

Combine -12x^{2} and -12x^{2} to get -24x^{2}.

$$6x^{3}-46x^{2}+106x-72+4x^{3}-24x^{2}+36x+8x-24$$

Combine 36x and 8x to get 44x.

$$6x^{3}-46x^{2}+106x-72+4x^{3}-24x^{2}+44x-24$$

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

$$10x^{3}-46x^{2}+106x-72-24x^{2}+44x-24$$

Combine -46x^{2} and -24x^{2} to get -70x^{2}.

$$10x^{3}-70x^{2}+106x-72+44x-24$$

Combine 106x and 44x to get 150x.

$$10x^{3}-70x^{2}+150x-72-24$$

Subtract 24 from -72 to get -96.

$$10x^{3}-70x^{2}+150x-96$$