To fill in the blanks for the polynomial equation that shows the subtraction of the second expression from the first, let's rewrite the expressions clearly:
First expression:
\[ 9x^2 + x + 13 \]
Second expression:
\[ 2x^2 + 12x + (x^2 + 5x + 2) \]
Now, we need to perform the subtraction:
\[ (9x^2 + x + 13) - (2x^2 + 12x + x^2 + 5x + 2) \]
First, combine the second expression:
\[ 2x^2 + 12x + x^2 + 5x + 2 = (2x^2 + x^2) + (12x + 5x) + 2 \] \[ = 3x^2 + 17x + 2 \]
Now, we will subtract this combined expression from the first expression:
\[ 9x^2 + x + 13 - (3x^2 + 17x + 2) \]
Distributing the negative sign gives:
\[ 9x^2 + x + 13 - 3x^2 - 17x - 2 \]
Now combine like terms:
- For \(x^2\) terms: \(9x^2 - 3x^2 = 6x^2\)
- For \(x\) terms: \(x - 17x = -16x\)
- For constant terms: \(13 - 2 = 11\)
Putting it all together, we have:
\[ 6x^2 - 16x + 11 \]
Therefore, the original subtraction problem, filled in completely, looks like this:
\[ 9x^2 + x + 13 - (2x^2 + 12x + (x^2 + 5x + 2)) = 6x^2 - 16x + 11 \]
So the filled-in blanks would be:
- In the second expression, the blank after \(+ 12x\) should be \(x^2\) (the start of the additional polynomial).
- In that additional polynomial, \(+ 5x\) should have the blank filled with \(2\) (the constant term).
Thus the complete answer is:
- Second expression blank: \(x^2\)
- Sum of that additional polynomial blank: \(2\)