Sure! Let's evaluate the polynomial \(-x^3 + 4x^2 - 3x + 12\) at \(x = -2\).
1. Start by substituting \(x = -2\) into the polynomial:
[tex]\[
-(-2)^3 + 4(-2)^2 - 3(-2) + 12
\][/tex]
2. Calculate \((-2)^3\):
[tex]\[
(-2)^3 = -8
\][/tex]
So, \(-(-2)^3\) becomes:
[tex]\[
-(-8) = 8
\][/tex]
3. Next, calculate \((-2)^2\):
[tex]\[
(-2)^2 = 4
\][/tex]
So, \(4(-2)^2\) becomes:
[tex]\[
4 \cdot 4 = 16
\][/tex]
4. Then calculate \(-3(-2)\):
[tex]\[
-3(-2) = 6
\][/tex]
5. Finally, combine all the terms together and add the constant term \(12\):
[tex]\[
8 + 16 + 6 + 12
\][/tex]
6. Sum these values step-by-step:
[tex]\[
8 + 16 = 24
\][/tex]
[tex]\[
24 + 6 = 30
\][/tex]
[tex]\[
30 + 12 = 42
\][/tex]
So, the value of the polynomial \(-x^3 + 4x^2 - 3x + 12\) at \(x = -2\) is:
[tex]\[
42
\][/tex]
