Factor -3x^6 + 12x^12 completely.

Step-by-step solution

1. 1

   Write the terms in descending order. Standard form makes the leading term positive and the structure visible:

   $12x^{12}-3x^6$

2. 2

   Extract the greatest common factor. The numerical GCF of $12$ and $3$ is $3$, and the smaller power of $x$ is $x^6$:

   $12x^{12}-3x^6=3x^6\left(4x^6-1\right)$

3. 3

   Do not stop here — inspect the bracket. $4x^6=\left(2x^3\right)^2$ and $1=1^2$, so $4x^6-1$ is a difference of squares. Leaving the answer as $3x^6(4x^6-1)$ is an incomplete factorisation.

4. 4

   Apply $A^2-B^2=(A-B)(A+B)$ with $A=2x^3$, $B=1$.

   $4x^6-1=\left(2x^3-1\right)\left(2x^3+1\right)$

   $12x^{12}-3x^6=3x^6\left(2x^3-1\right)\left(2x^3+1\right)$

5. 5

   Check that nothing more can be pulled out over the rationals. $2x^3\mp 1$ would need $2$ to be a rational cube to split as a sum or difference of cubes, and it is not. Verify numerically at $x=1$: original $=12-3=9$, and $3(1)(2-1)(2+1)=3\cdot 1\cdot 3=9$ $\checkmark$.