Question

solve the equation $12n^2 + 48n = -n^3 - 64$. the solution is $n = \square$.

Explanation:

Step1: Rearrange all terms to left

$$n^3 + 12n^2 + 48n + 64 = 0$$

Step2: Recognize perfect cube form

Notice that $n^3 + 12n^2 + 48n + 64$ matches $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$, where $a=n$, $b=4$:
$$(n+4)^3 = 0$$

Step3: Solve for n

Set the factor equal to 0:
$$n + 4 = 0$$

Answer:

$-4$