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

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

Evaluate

$$x\left(x+1\right)\left(x+6\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+2\right)^{3}.

$$x^{2}-2x+1+x^{3}+6x^{2}+12x+8-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine x^{2} and 6x^{2} to get 7x^{2}.

$$7x^{2}-2x+1+x^{3}+12x+8-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine -2x and 12x to get 10x.

$$7x^{2}+10x+1+x^{3}+8-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Add 1 and 8 to get 9.

$$7x^{2}+10x+9+x^{3}-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Use the distributive property to multiply x-2 by x+3 and combine like terms.

$$7x^{2}+10x+9+x^{3}-\left(x^{2}+x-6\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

To find the opposite of x^{2}+x-6, find the opposite of each term.

$$7x^{2}+10x+9+x^{3}-x^{2}-x+6+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}+10x+9+x^{3}-x+6+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine 10x and -x to get 9x.

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

Add 9 and 6 to get 15.

$$6x^{2}+9x+15+x^{3}+\left(x+4\right)\left(x-4\right)-3x+1$$

Consider \left(x+4\right)\left(x-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$6x^{2}+9x+15+x^{3}+x^{2}-16-3x+1$$

Combine 6x^{2} and x^{2} to get 7x^{2}.

$$7x^{2}+9x+15+x^{3}-16-3x+1$$

Subtract 16 from 15 to get -1.

$$7x^{2}+9x-1+x^{3}-3x+1$$

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

$$7x^{2}+6x-1+x^{3}+1$$

Add -1 and 1 to get 0.

$$7x^{2}+6x+x^{3}$$

Expand

$$x^{3}+7x^{2}+6x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+2\right)^{3}.

$$x^{2}-2x+1+x^{3}+6x^{2}+12x+8-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine x^{2} and 6x^{2} to get 7x^{2}.

$$7x^{2}-2x+1+x^{3}+12x+8-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine -2x and 12x to get 10x.

$$7x^{2}+10x+1+x^{3}+8-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Add 1 and 8 to get 9.

$$7x^{2}+10x+9+x^{3}-\left(x-2\right)\left(x+3\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

Use the distributive property to multiply x-2 by x+3 and combine like terms.

$$7x^{2}+10x+9+x^{3}-\left(x^{2}+x-6\right)+\left(x+4\right)\left(x-4\right)-3x+1$$

To find the opposite of x^{2}+x-6, find the opposite of each term.

$$7x^{2}+10x+9+x^{3}-x^{2}-x+6+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}+10x+9+x^{3}-x+6+\left(x+4\right)\left(x-4\right)-3x+1$$

Combine 10x and -x to get 9x.

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

Add 9 and 6 to get 15.

$$6x^{2}+9x+15+x^{3}+\left(x+4\right)\left(x-4\right)-3x+1$$

Consider \left(x+4\right)\left(x-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$6x^{2}+9x+15+x^{3}+x^{2}-16-3x+1$$

Combine 6x^{2} and x^{2} to get 7x^{2}.

$$7x^{2}+9x+15+x^{3}-16-3x+1$$

Subtract 16 from 15 to get -1.

$$7x^{2}+9x-1+x^{3}-3x+1$$

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

$$7x^{2}+6x-1+x^{3}+1$$

Add -1 and 1 to get 0.

$$7x^{2}+6x+x^{3}$$