Find x if (x + 4)(x^2 - 4x + 16) - x(x + 5)(x - 5) = 264.

Step-by-step solution

1. 1

   Recognise the first product as a sum of cubes. The pattern $a^3+b^3=(a+b)\left(a^2-ab+b^2\right)$ fits with $a=x$ and $b=4$, since $a^2-ab+b^2=x^2-4x+16$:

   $(x+4)\left(x^2-4x+16\right)=x^3+64$

   Expanding this by brute force also works but invites sign errors; recognising the identity makes the cancellation ahead visible.

2. 2

   Recognise the second product as a difference of squares.

   $(x+5)(x-5)=x^2-25\;\Longrightarrow\;x(x+5)(x-5)=x^3-25x$

3. 3

   Substitute both and watch the cubics cancel.

   $\left(x^3+64\right)-\left(x^3-25x\right)=25x+64$

   The $x^3$ terms disappear exactly, so what looked like a cubic equation is actually linear — that is the entire design of the problem.

4. 4

   Solve the linear equation.

   $25x+64=264\;\Longrightarrow\;25x=200\;\Longrightarrow\;x=8$

5. 5

   Check by substituting back into the original. With $x=8$: the first product is $12\left(64-32+16\right)=12\cdot 48=576$, and the second is $8\cdot 13\cdot 3=312$. Then

   $576-312=264\;\checkmark$

   The check is worth doing in full here, because the cancellation of the cubic terms is the kind of step that hides an arithmetic slip.