Factor (y - x)^2 (x + y)^2 - (x - y)^2 (x + y) completely.

Step-by-step solution

1. 1

   Make the two binomials match. The first term contains $(y-x)^{2}$ and the second $(x-y)^{2}$. Since $y-x=-(x-y)$ and squaring kills the sign,

   $(y-x)^{2}=(-(x-y))^{2}=(x-y)^{2}.$

   Rewriting the first term with $(x-y)^{2}$ makes the shared structure visible; without this step the common factor is easy to miss entirely.

2. 2

   Rewrite the expression in matched form. The problem becomes

   $(x-y)^{2}(x+y)^{2}-(x-y)^{2}(x+y).$

   Now both terms visibly contain $(x-y)^{2}$ and at least one copy of $(x+y)$.

3. 3

   Identify the greatest common factor. Comparing the two terms factor by factor: $(x-y)$ appears squared in both, and $(x+y)$ appears to the second power in the first term but only to the first power in the second. The GCF takes the lowest power of each:

   $\text{GCF}=(x-y)^{2}(x+y).$

4. 4

   Factor it out. Dividing each term by the GCF leaves $(x+y)$ from the first and $1$ from the second:

   $(x-y)^{2}(x+y)^{2}-(x-y)^{2}(x+y)=(x-y)^{2}(x+y)\left[(x+y)-1\right].$

   The $1$ is the part most often dropped — the second term does not vanish when the GCF is removed, it becomes $1$.

5. 5

   State the answer and verify numerically. The complete factorisation is

   $(x-y)^{2}(x+y)(x+y-1),$

   and none of the three factors breaks down further over the rationals. Checking at $x=2.4$, $y=1.1$: the original gives $14.7875$ and the factored form gives $14.7875$ ✓; a second random pair agrees to nine digits as well.