To solve the problem step-by-step, we need to determine the expression [tex]\( A^2 - (B + C) \)[/tex] in its simplest form. Let's analyze each component individually.
Given polynomials are:
[tex]\[ A = 3x - 4 \][/tex]
[tex]\[ B = x + 7 \][/tex]
[tex]\[ C = x^2 + 2 \][/tex]
1. Calculate [tex]\( A^2 \)[/tex]:
[tex]\[
A = 3x - 4
\][/tex]
[tex]\[
A^2 = (3x - 4)^2
\][/tex]
Expanding [tex]\( (3x - 4)^2 \)[/tex] using the binomial expansion formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(3x - 4)^2 = (3x)^2 - 2 \cdot 3x \cdot 4 + 4^2
\][/tex]
[tex]\[
= 9x^2 - 24x + 16
\][/tex]
2. Calculate [tex]\( B + C \)[/tex]:
[tex]\[
B = x + 7
\][/tex]
[tex]\[
C = x^2 + 2
\][/tex]
So, [tex]\( B + C \)[/tex] is:
[tex]\[
B + C = (x + 7) + (x^2 + 2)
\][/tex]
[tex]\[
= x^2 + x + 9
\][/tex]
3. Calculate [tex]\( A^2 - (B + C) \)[/tex]:
We now have:
[tex]\[
A^2 = 9x^2 - 24x + 16
\][/tex]
[tex]\[
B + C = x^2 + x + 9
\][/tex]
So:
[tex]\[
A^2 - (B + C) = (9x^2 - 24x + 16) - (x^2 + x + 9)
\][/tex]
Subtracting the polynomials:
[tex]\[
= 9x^2 - 24x + 16 - x^2 - x - 9
\][/tex]
Combine like terms:
[tex]\[
= (9x^2 - x^2) + (-24x - x) + (16 - 9)
\][/tex]
[tex]\[
= 8x^2 - 25x + 7
\][/tex]
Thus, the simplified form of [tex]\( A^2 - (B + C) \)[/tex] is:
[tex]\[
8x^2 - 25x + 7
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{8x^2 - 25x + 7} \][/tex]
