To complete the polynomial equation, we will subtract the second expression from the first and determine the resulting difference.
Given the first expression:
\( 9x^2 + x + 13 \)
And the second expression:
\( 2x^2 + 12x + (x^2 + 5x + 2) \)
First, we need to combine the terms of the second expression:
- Combine the \( x^2 \) terms:
- \( 2x^2 + x^2 = 3x^2 \)
- Combine the \( x \) terms:
- \( 12x + 5x = 17x \)
- The constant term remains \( 2 \).
Thus, the second expression simplifies to:
\( 3x^2 + 17x + 2 \)
Now, we can perform the subtraction:
\[
(9x^2 + x + 13) - (3x^2 + 17x + 2)
\]
Distributing the negative sign across the terms of the second expression:
\[
9x^2 + x + 13 - 3x^2 - 17x - 2
\]
Now, combine like terms.
1. For \( x^2 \) terms:
- \( 9x^2 - 3x^2 = 6x^2 \)
2. For \( x \) terms:
- \( x - 17x = -16x \)
3. For constant terms:
- \( 13 - 2 = 11 \)
Therefore, the resulting expression after subtracting is:
\[
6x^2 - 16x + 11
\]
Now we can fill in the blanks:
- The term representing the missing expression after subtraction is \( 3x^2 + 17x + 2 \).
- The resulting difference is \( 6x^2 - 16x + 11 \).
So the completed polynomial equation is:
\[
9x^2 + x + 13 - (3x^2 + 17x + 2) = 6x^2 - 16x + 11
\]