Simplify 8(x^2 - 2y^2) - x(7x + y) + xy and factor the result.

Step-by-step solution

1. 1

   Expand the first product.

   $8\left(x^2-2y^2\right)=8x^2-16y^2$

2. 2

   Expand the second product, keeping the leading minus attached.

   $-x(7x+y)=-7x^2-xy$

   Both terms inside get multiplied by $-x$; producing $-7x^2+xy$ here is the usual sign error.

3. 3

   Assemble everything and look for cancellation.

   $8x^2-16y^2-7x^2-xy+xy$

   The two $xy$ terms are exact opposites, so they vanish. That cancellation is clearly the point of the problem — it turns a mixed-term mess into a two-term expression.

4. 4

   Collect the surviving terms.

   $8x^2-7x^2=x^2\;\Longrightarrow\; x^2-16y^2$

5. 5

   Recognise and apply the difference-of-squares pattern. Since $16y^2=(4y)^2$,

   $x^2-16y^2=x^2-(4y)^2=(x-4y)(x+4y)$

   Checking at $x=3$, $y=1$: the original is $8(9-2)-3(21+1)+3=56-66+3=-7$, and the factored form gives $(3-4)(3+4)=-7\;\checkmark$.