If $m = -4$, $n = -2$, then the value of $m^3 - 3m^2 + 3m + 3n + 3n^2 + n^3$ is

## Concept & Formula This problem is designed to look like a tedious calculation, but it hides two standard algebraic cubic identities. The key formulas to recognize are: $$(x - 1)^3 = x^3 - 3x^2 + 3x - 1$$ $$(y + 1)^3 = y^3 + 3y^2 + 3y + 1$$ ## Step-by-Step Solution * **Given:** We need to evaluate the expression $m^3 - 3m^2 + 3m + 3n + 3n^2 + n^3$. * **Rearrangement:** Group the $m$ terms together and the $n$ terms together. $$(m^3 - 3m^2 + 3m) + (n^3 + 3n^2 + 3n)$$ * **Completing the Cubes:** Notice that the first group is exactly 1 unit away from being $(m-1)^3$, and the second group is exactly 1 unit away from being $(n+1)^3$. * Let's artificially add $-1$ and $+1$ to the expression to complete these identities without changing the overall value of the equation: $$(m^3 - 3m^2 + 3m - 1) + 1 + (n^3 + 3n^2 + 3n)$$ $$(m^3 - 3m^2 + 3m - 1) + (n^3 + 3n^2 + 3n + 1)$$ * **Substitution:** Compress the expression using the identities. $$(m - 1)^3 + (n + 1)^3$$ * **Calculation:** Now, substitute the given values $m = -4$ and $n = -2$ into this vastly simplified expression. $$(-4 - 1)^3 + (-2 + 1)^3$$ $$(-5)^3 + (-1)^3$$ $$-125 + (-1)$$ $$-126$$ ## Exam Strategy & Shortcut If you cannot spot the algebraic identities under pressure, do not panic. Because the given numbers are very small ($-4$ and $-2$), **direct substitution** is a perfectly viable and fast backup strategy. $m^3 = -64$ $-3m^2 = -3(16) = -48$ $3m = -12$ $n^3 = -8$ $3n^2 = 3(4) = 12$ $3n = -6$ Summing them up: $-64 - 48 - 12 - 8 + 12 - 6 = -126$. This takes about 30 seconds if you are careful with your negative signs. ## Common Pitfall When substituting negative values into powers, students frequently mess up the signs. Remember: A negative number to an odd power remains negative (e.g., $(-5)^3 = -125$). A negative number to an even power becomes positive (e.g., $(-4)^2 = +16$). Mixing these up during direct substitution will guarantee an incorrect answer. ## Final Answer Therefore, the correct answer is -126.