To determine the correct operation that results in the given simplified expression [tex]\( x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \)[/tex], we must analyze the polynomials [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] and perform operations step-by-step:
Given polynomials:
[tex]\[ P = x^4 + 3x^3 + 2x^2 - x + 2 \][/tex]
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
First, we need to expand the polynomial [tex]\( Q \)[/tex]:
[tex]\[ Q = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
[tex]\[ Q = x^3 \cdot x^2 + x^3 \cdot (-2) + 2x^2 \cdot x^2 + 2x^2 \cdot (-2) + 3 \cdot x^2 + 3 \cdot (-2) \][/tex]
[tex]\[ Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6 \][/tex]
[tex]\[ Q = x^5 + 2x^4 - 2x^3 - x^2 - 6 \][/tex]
The next step is to calculate the difference between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[ P - Q = (x^4 + 3x^3 + 2x^2 - x + 2) - (x^5 + 2x^4 - 2x^3 - x^2 - 6) \][/tex]
Distribute the negative sign:
[tex]\[ P - Q = x^4 + 3x^3 + 2x^2 - x + 2 - x^5 - 2x^4 + 2x^3 + x^2 + 6 = -x^5 + x^4 + 5x^3 + 3x^2 - x + 8 \][/tex]
However, we are looking for the simplified expression:
[tex]\[ x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]
We see that [tex]\(-P + Q\)[/tex] actually results in the given expression:
[tex]\[-x^5 - x^4 + 5x^3 + 3x^2 - x + 8 = x^5 + x^4 - 5x^3 - 3x^2 + x - 8.\][/tex]
This means there is a sign discrepancy and we actually calculated the opposite of what is intended by [tex]\(P-Q\)[/tex].
To conclude:
The correct answer is:
\[ \boxed{D} ]
