To fill in the blanks in the polynomial equation, we start with the following expression:
\[ \nonex + \text{blank} , x + 13 - (2x + 12x + \text{blank blank} , x^2 + 5x + 2) \]
First, let's simplify the subtraction of the second expression from the first. We will rewrite it as:
\[ \nonex + \text{blank} , x + 13 - 2x - 12x - \text{blank blank} , x^2 - 5x - 2 \]
Now, we can combine like terms:
-
For the \(x^2\) terms:
- There is a "\(- \text{blank blank} , x^2\)" term coming from the second expression, so we need to find out which term balances it out.
-
For the \(x\) terms:
- We have \(\text{blank} , x - 2x - 12x - 5x\). We will also denote the total coefficient for \(x\) which we will compute later.
-
For the constant terms:
- From the first expression we have +13, and from the second, we have -2. So the constant terms will be \(13 - 2 = 11\).
Next, we need to fill in the blanks systematically.
-
Let's denote the first term by \(a = \nonex\).
-
The coefficient of \(x\) in the first expression is blank, let's denote it \(b\).
Thus, our equation becomes:
\[ a + b , x + 13 - (2 + 12 + 5 + 2)x - \text{blank blank} , x^2 \]
Combining the \(x\) terms becomes:
\[ a + (b - 19)x + 11 - \text{blank blank} , x^2 \]
Thus, we equate the original polynomial with the simplified version while keeping structure. Since it mentions, “the difference”, we can deduce the total first expression must fulfill the structure of polynomial equation matching.
So, assuming we want our resulting expression to be balanced and in clear polynomial order:
This suggests:
- \(\text{blank blank} = 0\) so there will be no x squared term in the original.
- For \(b\), it looks to complete the balance.
Finally, rewriting:
We initially align \(a = 0\), \(b = 19\), \(\text{blank blank} = 0\).
Thus, filling the blanks, we can summarize as follows:
So, the completed expression looks something like:
\[ 0 + 19x + 13 - 2x - 12x - 0 , x^2 - 5x - 2 \]
Upon resolving, the blanks get filled as \(19\) for \(\text{blank} \), \(0\) for the representation of the x squared.
Therefore the complete equation comes aligned to deduce optimality towards structural polynomial qualities and balances out to be:
Result:
\[
\text{Answer: } 19 , (blank), 0 , (blank(blank))
\] for each respective polynomial coefficient laid out.
So \( \boxed{19} \) and \( \boxed{0} \) respectively would be correct deductions filling these blanks.