To solve for the missing polynomial in the given equation, follow these steps:
### Given Equation:
[tex]\[ -\left(20 - 4x - 5x^2\right) = 20 - 7x^2 \][/tex]
### Step-by-Step Solution:
1. Distribute the negative sign through the parentheses on the left-hand side:
[tex]\[
-(20 - 4x - 5x^2) = -20 + 4x + 5x^2
\][/tex]
Now, we rewrite the equation:
[tex]\[
-20 + 4x + 5x^2 = 20 - 7x^2
\][/tex]
2. Combine all terms on one side of the equation to simplify:
Add [tex]\(20 - 7x^2\)[/tex] to both sides of the equation to set it equal to zero:
[tex]\[
-20 + 20 + 4x + 5x^2 + 7x^2 = 0
\][/tex]
3. Combine like terms:
[tex]\[
0 + 4x + (5x^2 + 7x^2) = 0
\][/tex]
Simplify:
[tex]\[
4x + 12x^2 = 0
\][/tex]
4. Factor the polynomial expression:
Factor out the common term, [tex]\(x\)[/tex], from each term:
[tex]\[
4x - 12x^2 = 0
\][/tex]
### Match with Given Choices:
Given the solution's format matches one of the provided choices:
1. [tex]\(4x - 12x^2\)[/tex]
2. [tex]\(4x - 2x^2\)[/tex]
3. [tex]\(40 - 4x - 12x^2\)[/tex]
4. [tex]\(40 - 4x - 2x^2\)[/tex]
The answer is:
[tex]\[
\boxed{4x - 12x^2}
\][/tex]
So, the correct missing polynomial expression is:
[tex]\[
4x - 12x^2
\][/tex]
