Simplify 6x times (x - y) squared, plus 3 times (y - x) cubed.

Step-by-step solution

1. 1

   Relate the two brackets. They are negatives of each other: $y-x=-(x-y)$. This matters because raising to a power treats the sign differently depending on parity:

   $(y-x)^2=(x-y)^2\quad\text{but}\quad (y-x)^3=-(x-y)^3$

   An even exponent absorbs the minus; an odd one keeps it.

2. 2

   Rewrite everything in terms of $(x-y)$. Substituting the cube identity,

   $6x(x-y)^2+3(y-x)^3=6x(x-y)^2-3(x-y)^3$

   Now both terms are built from the same bracket, which is what makes factoring possible.

3. 3

   Identify and extract the common factor. The coefficients $6$ and $3$ share a factor $3$, and the lower power of the bracket is $(x-y)^2$, so the GCF is $3(x-y)^2$:

   $6x(x-y)^2-3(x-y)^3=3(x-y)^2\bigl[2x-(x-y)\bigr]$

4. 4

   Simplify what is left inside the brackets.

   $2x-(x-y)=2x-x+y=x+y$

   The subtraction flips both inner signs — writing $2x-x-y=x-y$ would be the standard error here.

5. 5

   State the result and verify.

   $6x(x-y)^2+3(y-x)^3=3(x-y)^2(x+y)$

   Check at $x=3$, $y=1$: the original is $6(3)(2)^2+3(-2)^3=72-24=48$, and the factored form gives $3(2)^2(4)=48$ ✓.