To solve for the missing polynomial, let's start by analyzing the given expression and the target expression:
1. We have the equation:
[tex]\[ -\left(20 - 4x - 5x^2 \right) = 20 - 7x^2. \][/tex]
2. First, distribute the negative sign through the left expression inside the parentheses:
[tex]\[ -20 + 4x + 5x^2. \][/tex]
3. Simplify the left expression:
[tex]\[ 5x^2 + 4x - 20. \][/tex]
Now we need to find which of the given options, when simplified, equals [tex]\( 5x^2 + 4x - 20 \)[/tex]:
- Option 1: [tex]\( 4x - 12x^2 \)[/tex]:
This does not match [tex]\( 5x^2 + 4x - 20 \)[/tex].
- Option 2: [tex]\( 4x - 2x^2 \)[/tex]:
This does not match [tex]\( 5x^2 + 4x - 20 \)[/tex].
- Option 3: [tex]\( 40 - 4x - 12x^2 \)[/tex]:
Simplify and compare:
[tex]\[ 40 - 4x - 12x^2 \neq 5x^2 + 4x - 20. \][/tex]
- Option 4: [tex]\( 40 - 4x - 2x^2 \)[/tex]:
Simplify and compare:
[tex]\[ 40 - 4x - 2x^2 \neq 5x^2 + 4x - 20. \][/tex]
4. None of the given expressions simplify to [tex]\(5x^2 + 4x - 20\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{\text{None of the given options are correct}} \][/tex]
