To solve the expression [tex]\(a^2 + 12abc - c^2\)[/tex] given the values [tex]\(a = -8\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = 7\)[/tex], let's go through the steps in detail.
1. Square of [tex]\(a\)[/tex]:
Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a = -8 \implies a^2 = (-8)^2 = 64
\][/tex]
2. Product of 12, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
Calculate [tex]\(12 \cdot a \cdot b \cdot c\)[/tex]:
[tex]\[
a = -8, \, b = 6, \, c = 7 \implies 12 \cdot (-8) \cdot 6 \cdot 7
\][/tex]
3. Calculation of the term [tex]\(12abc\)[/tex]:
First, calculate [tex]\( (-8) \cdot 6 = -48 \)[/tex].
Then, multiply the result by [tex]\( 7 \)[/tex]:
[tex]\[
-48 \cdot 7 = -336
\][/tex]
Finally, multiply by 12:
[tex]\[
12 \cdot (-336) = -4032
\][/tex]
4. Square of [tex]\(c\)[/tex]:
Calculate [tex]\(c^2\)[/tex]:
[tex]\[
c = 7 \implies c^2 = 7^2 = 49
\][/tex]
5. Combine all parts:
Substitute these values back into the expression:
[tex]\[
a^2 + 12abc - c^2 = 64 + (-4032) - 49
\][/tex]
6. Simplify the expression:
Combine [tex]\(64\)[/tex], [tex]\(-4032\)[/tex], and [tex]\(-49\)[/tex]:
[tex]\[
64 - 4032 - 49
\][/tex]
7. Final Calculation:
Performing the subtraction step-by-step:
[tex]\[
64 - 4032 = -3968
\][/tex]
[tex]\[
-3968 - 49 = -4017
\][/tex]
Therefore, the value of the expression [tex]\(a^2 + 12abc - c^2\)[/tex] for [tex]\(a = -8\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = 7\)[/tex] is [tex]\(-4017\)[/tex]. This matches our required answer.
Thus, the correct answer is:
[tex]\[
\boxed{-4017}
\][/tex]
