Factor $(x-3)^2-2(x-3)(y+2)-35(y+2)^2$

Question

My train of thought was first to look at it like a perfect square trinomial, but then I saw the $-35$ and had no idea. Can someone put me on the right track please?

Answer 1

Write $X=x-3$ and $Y=y+2$ then your expression reduces to $$X^2-2XY-35Y^2$$ which equals $$X^2-2XY+Y^2-36Y^2 \\=(X-Y)^2-36Y^2\\=(X-Y+6Y)(X-Y-6Y)\\=(X+5Y)(X-7Y)$$

Now substitute the values of $X$ and $Y$ to get the required factors of your given expression.