Find the Missing Term in (3x-8)(-4+?) = -3x²-4x+32

Complete the missing element

$(3x-8)(-4+?)=-3x^2-4x+32$

To solve this problem, we'll follow these steps:

• Step 1: Expand the binomial expression using the distributive property.
• Step 2: Compare the resulting polynomial's coefficients with the given expression's coefficients.
• Step 3: Solve for the missing element.

Now, let's work through each step:
Step 1: Express $(3x-8)(-4+?)$ using the distributive property:
$(3x)(-4) + (3x)(?) - (8)(-4) - (8)(?)$.
This simplifies to: $-12x + 3x(?) + 32 - 8(?)$.

Step 2: Compare with $-3x^2 - 4x + 32$, equaling terms by degree:
- The constant term (32) already matches.
- The $x$ term is $-12x + 3x(?) = -4x$.
- The $x^2$ term arises from $3x(?)$.

Step 3: Solve for the missing element by aligning coefficients:
$3x(?) = -3x^2$, therefore $? = -x$.
Thus, the missing element is $-x$.

Therefore, the solution to the problem is $-x$, which corresponds to choice 2.