Final Answer:
The solution for n is n = 0.
Step-by-step explanation:
Factor the quadratic inside the parentheses:
-n^2(n^2 + 5n + 6) = 0
-n^2(n + 3)(n + 2) = 0
Apply the zero product property:
If a * b * c = 0, then a = 0, b = 0, or c = 0.
In this case, we have:
-n^2 * (n + 3) * (n + 2) = 0
Therefore, one or more of the following must be true:
n^2 = 0
n + 3 = 0
n + 2 = 0
Solve for each possible value of n:
n^2 = 0: n = 0
n + 3 = 0: n = -3
n + 2 = 0: n = -2
Verify the solutions:
Substitute each solution back into the original equation to ensure it equals zero.
n = 0: -0^2(0^2 + 5(0) + 6) = 0 (True)
n = -3: -(-3)^2((-3)^2 + 5(-3) + 6) = 0 (False)
n = -2: -(-2)^2((-2)^2 + 5(-2) + 6) = 0 (False)
Therefore, the only valid solution for n is:
n = 0