Question

simplify the expression $3x(x - 12x)+3x^{2}-2(x - 2)^{2}$. which statements are true about the process and simplified product? choose three correct answers.
the term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$.
the parentheses are eliminated through multiplication.
after multiplying, the like terms are combined by adding and subtracting.
the simplified product is a binomial.
the final simplified product is $-28x^{2}+8x - 8$

Explanation:

Step1: Expand $3x(x-12x)$

$3x(x-12x) = 3x(-11x) = -33x^2$

Step2: Expand $-2(x-2)^2$

First square $(x-2)$: $(x-2)^2 = x^2 -4x +4$, then multiply by -2: $-2(x^2 -4x +4) = -2x^2 +8x -8$

Step3: Combine all terms

Bring in the remaining $+3x^2$, so total terms: $-33x^2 +3x^2 -2x^2 +8x -8$

Step4: Combine like terms

$$\begin{align*} (-33+3-2)x^2 +8x -8 &= -32x^2 +8x -8 \end{align*}$$

Step5: Evaluate each statement

1. Check $-2(x-2)^2$ simplification: Squaring $(x-2)$ first is correct.
2. Check parentheses elimination: Multiplication removes parentheses, correct.
3. Check like terms: Combined via addition/subtraction, correct.
4. Check if result is binomial: Result has 3 terms, so false.
5. Check final product: Our result is $-32x^2+8x-8$, not $-28x^2+8x-8$, so false.

Answer:

A. The term $-2(x - 2)^2$ is simplified by first squaring the expression $x - 2$.
B. The parentheses are eliminated through multiplication.
C. After multiplying, the like terms are combined by adding and subtracting.