To determine the missing polynomial, let’s solve the given equation step-by-step:
We start with the equation:
[tex]\[ ? - (20 - 4x - 5x^2) = 20 - 7x^2 \][/tex]
Let's first simplify the left-hand side (LHS) of the equation by distributing the negative sign within the parentheses.
[tex]\[ ? - 20 + 4x + 5x^2 = 20 - 7x^2 \][/tex]
To isolate the missing polynomial (denoted as "?"), we need to move the constant term and the polynomial terms from the LHS to the right-hand side (RHS).
To do this, add 20, subtract 4x, and subtract 5x^2 from both sides:
[tex]\[ ? = 20 - 7x^2 + 20 - 4x - 5x^2 \][/tex]
Combine the like terms on the RHS:
1. Combine the constant terms:
[tex]\[ 20 + 20 = 40 \][/tex]
2. Combine the [tex]\(x\)[/tex] terms:
[tex]\[ 0x - 4x = -4x \][/tex]
3. Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[ -7x^2 - 5x^2 = -12x^2 \][/tex]
So, the resulting polynomial on the right-hand side is:
[tex]\[ 40 - 4x - 12x^2 \][/tex]
Therefore, the missing polynomial is:
[tex]\[ 40 - 4x - 12x^2 \][/tex]
This matches one of the provided answer choices: [tex]\( \boxed{40 - 4x - 12x^2} \)[/tex].
