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

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

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

Answer

Explanation:

Step1: Simplify each part

  • For (3x(x - 12x)), first simplify inside the parentheses: (x-12x=-11x), then (3x\times(-11x)=-33x^{2}).
  • For (-2(x - 2)^{2}), use the formula ((a - b)^{2}=a^{2}-2ab + b^{2}) (so we first square (x - 2)). Here (a = x) and (b = 2), so ((x - 2)^{2}=x^{2}-4x + 4), and (-2(x^{2}-4x + 4)=-2x^{2}+8x-8).
  • The original expression (3x(x - 12x)+3x^{2}-2(x - 2)^{2}) becomes (-33x^{2}+3x^{2}-2x^{2}+8x - 8).

Step2: Combine like - terms

Combine the (x^{2}) terms: (-33x^{2}+3x^{2}-2x^{2}=(-33 + 3-2)x^{2}=-32x^{2}). The expression is (-32x^{2}+8x - 8) (a trinomial). But let's check each option:

  • Option 1: The term (-2(x - 2)^{2}) is simplified by first squaring (x - 2) (using ((a - b)^{2}=a^{2}-2ab + b^{2})), this is True.
  • Option 2: The simplified product (-32x^{2}+8x - 8) is a trinomial (not a binomial), so this is False.
  • Option 3: After multiplying (expanding each part), we combine like terms (e.g., combining (x^{2}) terms) by adding and subtracting. This is True.
  • Option 4: We use the distributive property (a(b + c)=ab+ac) (for (3x(x - 12x)) and (-2(x - 2)^{2}=-2(x^{2}-4x + 4))) to eliminate parentheses. This is True.
  • Option 5: The final simplified product is (-32x^{2}+8x - 8\neq-28x^{2}+8x - 8). This is False.

Answer:

The first, third, and fourth options are True. So the correct options 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.