To find the missing polynomial in the equation
[tex]\[ ? - \left(20 - 4x - 5x^2\right) = 20 - 7x^2, \][/tex]
we can follow these steps:
1. Expand and simplify the equation:
[tex]\[ ? - 20 + 4x + 5x^2 = 20 - 7x^2 \][/tex]
2. Isolate the missing polynomial(?):
Shift all terms involving \('?'\) to one side and the constants and other terms to the other side. This yields:
[tex]\[ ? = 20 - 7x^2 + 20 - 4x + 5x^2 \][/tex]
3. Combine like terms:
- Constant terms: \(20 + 20 = 40\)
- Linear terms: \(-4x\)
- Quadratic terms: \(5x^2 - 7x^2 = -2x^2\)
Therefore, combining all like terms gives us the polynomial:
[tex]\[ ? = 40 - 4x - 2x^2 \][/tex]
4. Compare to provided options to find the correct polynomial:
The options given are:
- \(4 x - 12 x^2\)
- \(4 x - 2 x^2\)
- \(40 - 4 x - 12 x^2\)
- \(40 - 4 x - 2 x^2\)
From these, the correct polynomial is:
[tex]\[ 40 - 4 x - 2 x^2 \][/tex]
Thus, the missing polynomial is [tex]\(\boxed{40 - 4x - 2x^2}\)[/tex].
