I have an exercise of factoring:
$7pqy^2 - 5y^2 - 5x^2 + 7pqx^2$, i must factor it.
I made this:
$7pqx^2 + 7pqy^2 -5x^2 - 5y^2$
= $-7pq(-x^2 - y^2) +5(-x^2 - y^2)$
= $(-x^2 - y^2)(5 - 7pq)$
= $(-x -y)(-x +y)(5-7pq)$
But the correct result should be:
$(x^2 +y^2)(-5 +7pq)$, that is not equal nor equivalent.
I also arrive at this result, factoring by $7pq$ instead of $-7pq$
So I do not want to know how to get to the result, but to know what I'm wrong about doing in this way.
On
In your last step, you incorrectly treated $(-x^2 - y^2)$ as difference of squares.
Difference of squares would apply if you had $(x^2 - y^2)$ as a factor, but you don't. What you have as a factor is $(-x^2 - y^2) = -(x^2 + y^2)$.
$$(-x^2 - y^2)(5 - 7pq) = -(x^2 + y^2)(5-7pq) = (x^2 + y^2)(7pq - 5)$$
You went wrong in the following step:
$$-x^2-y^2=-(x^2+y^2)\ne(-x-y)(-x+y)$$