To find the value of the polynomial \( n^3 + 5n^2 + 5n - 2 \) at \( n = -2 \), we will evaluate each term of the polynomial individually and then sum them up.
1. Evaluate \( n^3 \):
[tex]\[
n^3 = (-2)^3 = -8
\][/tex]
2. Evaluate \( 5n^2 \):
[tex]\[
5n^2 = 5 \times (-2)^2 = 5 \times 4 = 20
\][/tex]
3. Evaluate \( 5n \):
[tex]\[
5n = 5 \times (-2) = -10
\][/tex]
4. The constant term \( -2 \) remains unchanged:
[tex]\[
-2
\][/tex]
Now, add all these evaluated terms together:
[tex]\[
n^3 + 5n^2 + 5n - 2 = -8 + 20 - 10 - 2
\][/tex]
Let's perform the addition step by step:
[tex]\[
-8 + 20 = 12
\][/tex]
[tex]\[
12 - 10 = 2
\][/tex]
[tex]\[
2 - 2 = 0
\][/tex]
Therefore, the value of the polynomial \( n^3 + 5n^2 + 5n - 2 \) when \( n = -2 \) is \( 0 \).
Hence, the final answer is:
[tex]\[
\boxed{0}
\][/tex]
