To determine which polynomial expression represents a sum of cubes, we need to recall the formula for the sum of cubes, which is given by:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Let's analyze each given option to see if any of them match this formula.
1. Option 1: [tex]\((6-s)(s^2 + 6s + 36)\)[/tex]
Let's expand this expression:
[tex]\[
(6 - s)(s^2 + 6s + 36) = 6s^2 + 36s + 216 - s^3 - 6s^2 - 36s
\][/tex]
Combining like terms:
[tex]\[
0 - s^3 + 0 + 216 = -s^3 + 216
\][/tex]
This does not represent a sum of cubes.
2. Option 2: [tex]\((6 + s)(s^2 - 6s - 36)\)[/tex]
Let's expand this expression:
[tex]\[
(6 + s)(s^2 - 6s - 36) = 6s^2 - 36s - 216 + s^3 - 6s^2 - 36s
\][/tex]
Combining like terms:
[tex]\[
s^3 + 0 - 72s - 216
\][/tex]
This does not represent a sum of cubes.
3. Option 3: [tex]\((6 + s)(s^2 - 6s + 36)\)[/tex]
Let's expand this expression:
[tex]\[
(6 + s)(s^2 - 6s + 36) = 6s^2 - 36s + 216 + s^3 - 6s^2 + 36s
\][/tex]
Combining like terms:
[tex]\[
s^3 + 0 + 0 + 216 = s^3 + 216
\][/tex]
This represents [tex]\(s^3 + 6^3\)[/tex], which is indeed a sum of cubes. So, this option matches the sum of cubes formula when [tex]\(a = s\)[/tex] and [tex]\(b = 6\)[/tex].
4. Option 4: [tex]\((6 + s)(s^2 + 6s + 36)\)[/tex]
Let's expand this expression:
[tex]\[
(6 + s)(s^2 + 6s + 36) = 6s^2 + 36s + 216 + s^3 + 6s^2 + 36s
\][/tex]
Combining like terms:
[tex]\[
s^3 + 12s^2 + 72s + 216
\][/tex]
This does not represent a sum of cubes.
Given our analysis, the polynomial expression that represents a sum of cubes is:
[tex]\[ \boxed{(6 + s)(s^2 - 6s + 36)} \][/tex]
