Simplify (x + 2y)² − (x − 2y)².

Step-by-step solution

1. 1

   Recognise the difference-of-squares form. Two perfect squares separated by a minus sign always factor as

   $A^2 - B^2 = (A-B)(A+B)$

   With two variables this is far safer than expanding, where the $x^2$ and $4y^2$ terms give plenty of chances to lose a sign.

2. 2

   Name the bases.

   $A = x + 2y, \qquad B = x - 2y$

   The two bases differ only in the sign of the $2y$ term, so one of the factors below will contain only $y$ and the other only $x$.

3. 3

   Compute A − B.

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

   The subtraction of $-2y$ becomes $+2y$, doubling the $y$ term rather than cancelling it.

4. 4

   Compute A + B and multiply the factors.

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

   $(x+2y)^2 - (x-2y)^2 = 4y \cdot 2x = 8xy$

5. 5

   Verify by expansion and a numeric spot check. Expanding directly:

   $(x^2 + 4xy + 4y^2) - (x^2 - 4xy + 4y^2) = 8xy$

   Only the cross terms survive. With $x = 3$, $y = 2$: $(3+4)^2 - (3-4)^2 = 49 - 1 = 48$ and $8 \cdot 3 \cdot 2 = 48$.