Simplify (x + y)(x - y) + 4(y - 1), and factor the result.

Step-by-step solution

1. 1

   Expand the product with the difference-of-squares identity.

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

   Multiplying term by term would give the same thing, but the identity skips the two cross terms that cancel.

2. 2

   Expand the second bracket.

   $4(y-1) = 4y - 4$

3. 3

   Combine into the simplified expression.

   $x^2 - y^2 + 4y - 4$

   No two of these terms are alike, so this is the fully simplified expanded form.

4. 4

   Spot a second difference of squares. Grouping the last three terms and factoring out $-1$:

   $-y^2 + 4y - 4 = -\left(y^2 - 4y + 4\right) = -(y-2)^2$

   so the whole expression is $x^2 - (y-2)^2$ — again a difference of two squares.

5. 5

   Factor.

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

6. 6

   Check numerically at (x, y) = (3, 5). The original is $(8)(-2) + 4(4) = -16 + 16 = 0$. The expanded form gives $9 - 25 + 20 - 4 = 0$. The factored form gives $(3-5+2)(3+5-2) = 0 \times 6 = 0$. All three agree, and the zero is no accident: $x = y - 2$ makes the first factor vanish.