simplify the expression 3x(x - 12x)+3x^{2}-2(x - 2)^{2}. which statements are true about the process and…

Explanation:

Step1: Expand expressions

First, expand $3x(x - 12x)=3x\times x-3x\times12x = 3x^{2}-36x^{2}$. Expand $-2(x - 2)^{2}=-2(x^{2}-4x + 4)=-2x^{2}+8x - 8$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$. The original expression $3x(x - 12x)+3x^{2}-2(x - 2)^{2}$ becomes $3x^{2}-36x^{2}+3x^{2}-2x^{2}+8x - 8$.

Step2: Combine like - terms

Combine the $x^{2}$ terms: $(3 - 36+3 - 2)x^{2}+8x - 8=-32x^{2}+8x - 8$.

Now let's check the statements:

• For the term $-2(x - 2)^{2}$, we do first square the expression $x - 2$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$, so the statement "The term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$" is true.
• The simplified product $-32x^{2}+8x - 8$ is a trinomial, not a binomial, so the statement "The simplified product is a binomial" is false.
• After multiplying out the expressions, we combine like - terms by adding and subtracting the coefficients of the same degree terms, so the statement "After multiplying, the like terms are combined by adding and subtracting" is true.
• We use the distributive property $a(b + c)=ab+ac$ to eliminate the parentheses through multiplication, so the statement "The parentheses are eliminated through multiplication" is true.
• The final simplified product is $-32x^{2}+8x - 8$, not $-28x^{2}+8x - 8$, so the statement "The final simplified product is $-28x^{2}+8x - 8$" is false.

Answer:

The true statements are: The term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$. After multiplying, the like terms are combined by adding and subtracting. The parentheses are eliminated through multiplication.