Simplify (x + y)(x - y) + y(y - 2).

Step-by-step solution

1. 1

   Read the square brackets as ordinary grouping. $[\ ]$ and $(\ )$ mean the same thing here, so the expression is $(x+y)(x-y)+y(y-2)$: a product of two binomials plus a monomial times a binomial.

2. 2

   Expand the first product with the difference of squares. The two brackets are conjugates, so

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

   Using the identity beats FOIL because the cross terms $+xy$ and $-xy$ are guaranteed to cancel.

3. 3

   Expand the second product by distributing $y$.

   $y(y-2)=y^2-2y$

4. 4

   Add the two results and watch the $y^2$ terms annihilate.

   $x^2-y^2+y^2-2y=x^2-2y$

   The $-y^2$ from the identity exactly offsets the $+y^2$ from the distribution — the whole point of the problem.

5. 5

   Check numerically. At $x=3$, $y=2$: the original is $(5)(1)+2(0)=5$, and $x^2-2y=9-4=5$ ✓. At $x=1$, $y=4$: the original is $(5)(-3)+4(2)=-15+8=-7$, and $1-8=-7$ ✓.