Simplify (x + 3)² + (x − 2)² − 2(x + 3)(x − 2).

Step-by-step solution

1. 1

   Match the three terms to a known identity. The layout is a square, plus another square, minus twice the product of the two bases. That is

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

   Use it in reverse: the expression is already the expanded form of a single square.

2. 2

   Assign the two bases. Here

   $A = x+3, \qquad B = x-2$

   Check the middle term: $2AB = 2(x+3)(x-2)$, which is exactly what is being subtracted, so the identity applies with no adjustment.

3. 3

   Compute A − B, watching the double negative.

   $A - B = (x+3) - (x-2) = x + 3 - x + 2 = 5$

   The $x$ terms cancel because both bases have leading coefficient $1$ — this is why the answer will not depend on $x$ at all.

4. 4

   Square the result.

   $(A-B)^2 = 5^2 = 25$

   So the expression is the constant $25$ for every value of $x$.

5. 5

   Confirm by brute-force expansion. Term by term:

   $(x^2 + 6x + 9) + (x^2 - 4x + 4) - 2(x^2 + x - 6)$

   $= 2x^2 + 2x + 13 - 2x^2 - 2x + 12 = 25$

   Both the $x^2$ and the $x$ columns vanish, confirming the constant. Testing $x = 0$ gives $9 + 4 - 2(-6) = 25$ as well.