Let's break down the problem and determine which operation on the polynomials [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] results in the given simplified expression.
First, let's rewrite the polynomials:
[tex]\[ P(x) = x^4 + 3x^3 + 2x^2 - x + 2 \][/tex]
[tex]\[ Q(x) = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
Next, we'll expand [tex]\( Q(x) \)[/tex]:
[tex]\[ Q(x) = (x^3 + 2x^2 + 3)(x^2 - 2) \][/tex]
[tex]\[ Q(x) = x^3 \cdot x^2 + x^3 \cdot (-2) + 2x^2 \cdot x^2 + 2x^2 \cdot (-2) + 3 \cdot x^2 + 3 \cdot (-2) \][/tex]
[tex]\[ Q(x) = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6 \][/tex]
[tex]\[ Q(x) = x^5 + 2x^4 - 2x^3 - x^2 - 6 \][/tex]
Now let's examine the given simplified expression:
[tex]\[ x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]
We need to compare this expression against the results of performing various operations on [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].
Let's consider each operation one by one.
Operation: [tex]\( P(x) - Q(x) \)[/tex]
[tex]\[ P(x) - Q(x) = \left( x^4 + 3x^3 + 2x^2 - x + 2 \right) - \left( x^5 + 2x^4 - 2x^3 - x^2 - 6 \right) \][/tex]
[tex]\[ P(x) - Q(x) = -x^5 + x^4 + 5x^3 + 3x^2 - x + 8 \][/tex]
Operation: [tex]\( Q(x) - P(x) \)[/tex]
[tex]\[ Q(x) - P(x) = \left( x^5 + 2x^4 - 2x^3 - x^2 - 6 \right) - \left( x^4 + 3x^3 + 2x^2 - x + 2 \right) \][/tex]
[tex]\[ Q(x) - P(x) = x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]
Given the comparison, it is clear that the correct operation is:
[tex]\[ Q(x) - P(x) \][/tex]
Thus, the operation that results in the simplified expression:
[tex]\[ x^5 + x^4 - 5x^3 - 3x^2 + x - 8 \][/tex]
is
[tex]\[ \boxed{Q - P} \][/tex]
The answer is:
B. [tex]\( Q - P \)[/tex]
