Solve: (x-1)(x+1)(x-2) = -2x²-x³ | Comparing Factored and Expanded Forms

Solve the following problem:

$(x-1)(x+1)(x-2)=-2x^2-x^3$

Solve the equation by simplifying the expression on the left side in two steps. First, we'll proceed to multiply the expressions in the two leftmost pairs of parentheses:

Apply the shortened multiplication formula for squaring a binomial:

$(a\pm b)^2=a^2\pm2ab+b^2$

Due to the fact that these two pairs of parentheses are being multiplied by another expression (which is also in parentheses), place the result inside of parentheses (marked with an underline):

$\underline{ (x-1)(x+1)}(x-2)=-2x^2+x^3 \\ \underline{ (x^2-1^2)}(x-2)=-2x^2+x^3 \\ (x^2-1)(x-2)=-2x^2+x^3$

Continue to simplify the expression on the left side by using the expanded distribution law:

$(a+b)(c+d)=ac+ad+bc+bd$

Additionally, apply the law of exponents for multiplying terms with equal bases:

$a^ma^n=a^{m+n}$ We'll now apply these laws and expand the parentheses in the expression in the equation:

$(x^2-1)(x-2)=-2x^2+x^3 \\ x^3-2x^2-x+2=-2x^2+x^3 \\$Continue to combine like terms, by moving terms between sides. Later - we can see that the terms with squared and cubed powers cancel out, therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^3-2x^2-x+2=-2x^2+x^3\\ -x=-2\hspace{8pt}\text{/}:(-1)\\ \boxed{x=2}$

Therefore, the correct answer is answer C.