To simplify the expression:
\[ (6x^3 + 2x^2 - 5x - 14) - (-5x^3 + 7x^2 + 2x - 1)(6x^3 + 2x^2 - 5x - 14) - (-5x^3 + 7x^2 + 2x - 1) \]
we first need to distribute the negative signs and handle the subtraction correctly.
Given the first part of the expression as \( (6x^3 + 2x^2 - 5x - 14) \), we then have:
- The second part, which is the negative of a negative polynomial, becomes positive:
\[
- (-5x^3 + 7x^2 + 2x - 1) = 5x^3 - 7x^2 - 2x + 1 \]
So far, we have:
\[ (6x^3 + 2x^2 - 5x - 14) + (5x^3 - 7x^2 - 2x + 1) \]
Next, we subtract the second term from the first part. The full expression simplifies as follows:
\[ (6x^3 + 2x^2 - 5x - 14) - (5x^3 - 7x^2 - 2x + 1) \]
Remember that distributing the negative sign will reverse the signs of the second polynomial.
Thus, the expression becomes:
\[ (6x^3 + 2x^2 - 5x - 14) - 5x^3 + 7x^2 + 2x - 1 \]
Combining all terms:
- Combine \( 6x^3 - 5x^3 = 1x^3 \)
- Combine \( 2x^2 + 7x^2 = 9x^2 \)
- Combine \( -5x + 2x = -3x \)
- Combine \( -14 - 1 = -15 \)
The simplified expression will be:
\[ x^3 + 9x^2 - 3x - 15 \]
Now filling in the blanks in your format:
- The first polynomial: \( \underline{+}6x^3\), \( \underline{+}2x^2\), \( \underline{-}5x\), \( \underline{-}14\)
- The second polynomial (with the signs changed because of the negative sign in front): \( \underline{+}5x^3\), \( \underline{-}7x^2\), \( \underline{-}2x\), \( \underline{+}1\)
So summarized, the filled blanks with correct signs for the two expressions is:
( 6x^3 \quad 2x^2 \quad -5x \quad -14 \quad +5x^3 \quad -7x^2 \quad -2x \quad +1 )