If ` a + b + c = - 11 ` then what is the value of ` ( a + 4) ^(3) + ( b + 5) ^(3) + ( c + 2) ^(3) - 3 ( a + 4) ( b + 5) ( c + 5)` ?

Options

A. `-1331`
B. `-121`
C. 0
D. 1331

Answer

To solve the problem, we need to evaluate the expression:

\[ (a + 4)^3 + (b + 5)^3 + (c + 2)^3 - 3(a + 4)(b + 5)(c + 2) \]

Given that \( a + b + c = -11 \), we can rearrange this equation to express it in a form that will help us apply the identity for the sum of cubes:

\[ a + b + c + 11 = 0 \]

Now, we can use the identity for the sum of cubes:

\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) \]

In our case, we can define:

- \( x = a + 4 \)
- \( y = b + 5 \)
- \( z = c + 2 \)

Now, we need to find \( x + y + z \):

\[ x + y + z = (a + 4) + (b + 5) + (c + 2) = a + b + c + 11 \]

Since we know \( a + b + c = -11 \):

\[ x + y + z = -11 + 11 = 0 \]

Now, since \( x + y + z = 0 \), we can apply the identity:

\[ x^3 + y^3 + z^3 - 3xyz = 0 \]

Thus, we have:

\[ (a + 4)^3 + (b + 5)^3 + (c + 2)^3 - 3(a + 4)(b + 5)(c + 2) = 0 \]

Therefore, the value of the expression is:

\[ \boxed{0} \]