Let's break down the problem step-by-step given the polynomials [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex]:
1. Define the polynomials:
[tex]\[
A = x^2
\][/tex]
[tex]\[
B = 3x + 2
\][/tex]
[tex]\[
C = x - 3
\][/tex]
2. Compute [tex]\(AB\)[/tex]:
[tex]\[
AB = A \cdot B = x^2 \cdot (3x + 2)
\][/tex]
We distribute [tex]\(x^2\)[/tex] across each term inside the parentheses:
[tex]\[
AB = x^2 \cdot 3x + x^2 \cdot 2 = 3x^3 + 2x^2
\][/tex]
3. Compute [tex]\(C^2\)[/tex]:
[tex]\[
C^2 = (x - 3)^2
\][/tex]
We use the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
C^2 = x^2 - 2 \cdot x \cdot 3 + 3^2 = x^2 - 6x + 9
\][/tex]
4. Subtract [tex]\(C^2\)[/tex] from [tex]\(AB\)[/tex]:
[tex]\[
AB - C^2 = (3x^3 + 2x^2) - (x^2 - 6x + 9)
\][/tex]
Distribute the negative sign and combine like terms:
[tex]\[
AB - C^2 = 3x^3 + 2x^2 - x^2 + 6x - 9
\][/tex]
Simplify:
[tex]\[
AB - C^2 = 3x^3 + (2x^2 - x^2) + 6x - 9 = 3x^3 + x^2 + 6x - 9
\][/tex]
So, the expression [tex]\(AB - C^2\)[/tex] in simplest form is:
[tex]\[
3x^3 + x^2 + 6x - 9
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{3 x^3 + x^2 + 6 x - 9}
\][/tex]
