To fill in the blanks in the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference, we start with the expressions you provided:
1. **First Expression:** \( 9x^2 + bx + 13 \) (we need to find \( b \)).
2. **Second Expression:** \( - (2x^2 + 12x + c) \) (we need to find \( c \)).
3. **Resulting Difference:** \( dx^2 + 5x + 2 \) (we need to find \( d \) and confirm the coefficient for \( x \)).
### Step 1: Write the subtraction equation
\[
(9x^2 + bx + 13) - (2x^2 + 12x + c) = dx^2 + 5x + 2
\]
### Step 2: Perform the operation
First, distribute the negative sign:
\[
9x^2 + bx + 13 - 2x^2 - 12x - c = dx^2 + 5x + 2
\]
### Step 3: Combine like terms
Combine the \( x^2 \), \( x \), and constant terms on the left side:
- \( x^2 \) terms: \( 9x^2 - 2x^2 = 7x^2 \)
- \( x \) terms: \( bx - 12x \)
- Constant terms: \( 13 - c \)
The equation now looks like:
\[
(7x^2 + (b - 12)x + (13 - c)) = dx^2 + 5x + 2
\]
### Step 4: Equate coefficients
Now we can set up equations based on the coefficients:
1. For \( x^2 \): \( 7 = d \)
2. For \( x \): \( b - 12 = 5 \)
3. For the constant: \( 13 - c = 2 \)
### Step 5: Solve for \( b \) and \( c \)
- From \( b - 12 = 5 \):
\[
b = 5 + 12 = 17
\]
- From \( 13 - c = 2 \):
\[
c = 13 - 2 = 11
\]
### Step 6: Fill in the blanks
Thus, the filled polynomial equation is:
\[
9x^2 + (17)x + 13 - (2x^2 + 12x + 11) = (7)x^2 + 5x + 2
\]
Final blanks filled:
- \( () = 17 \)
- \( () = 11 \)
- \( () = 7 \)
The complete equation is:
\[
9x^2 + \mathbf{17}x + 13 - (2x^2 + 12x + \mathbf{11}) = \mathbf{7}x^2 + 5x + 2
\]