Let's break down the problem step by step to find what [tex]\( A B - C^2 \)[/tex] simplifies to.
Given the polynomials:
[tex]\[ A = x^2 \][/tex]
[tex]\[ B = 3x + 2 \][/tex]
[tex]\[ C = x - 3 \][/tex]
First, we need to compute the product [tex]\( AB \)[/tex]:
[tex]\[ AB = (x^2) \cdot (3x + 2) \][/tex]
Multiplying these, we get:
[tex]\[ AB = x^2 \cdot 3x + x^2 \cdot 2 = 3x^3 + 2x^2 \][/tex]
Next, we calculate [tex]\( C^2 \)[/tex]:
[tex]\[ C^2 = (x - 3)^2 \][/tex]
Expanding this, we have:
[tex]\[ C^2 = (x - 3)(x - 3) = x^2 - 6x + 9 \][/tex]
Now, to find [tex]\( A B - C^2 \)[/tex], we subtract [tex]\( C^2 \)[/tex] from [tex]\( AB \)[/tex]:
[tex]\[ AB - C^2 = (3x^3 + 2x^2) - (x^2 - 6x + 9) \][/tex]
Distributing the negative sign across the terms in [tex]\( C^2 \)[/tex], we get:
[tex]\[ AB - C^2 = 3x^3 + 2x^2 - x^2 + 6x - 9 \][/tex]
Combining like terms, we obtain:
[tex]\[ AB - C^2 = 3x^3 + (2x^2 - x^2) + 6x - 9 \][/tex]
[tex]\[ AB - C^2 = 3x^3 + x^2 + 6x - 9 \][/tex]
Thus, the simplified form of [tex]\( AB - C^2 \)[/tex] is:
[tex]\[ 3x^3 + x^2 + 6x - 9 \][/tex]
From the given answer choices, the correct one is:
[tex]\[ \boxed{3 x^3 + x^2 + 6 x - 9} \][/tex]
