Simplify (2x + 1)² − (2x − 1)².

Step-by-step solution

1. 1

   Choose the factoring route, not the expanding route. A difference of two squares always factors:

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

   Here both terms are already squares, so this identity turns a quadratic-looking problem into two short subtractions and one multiplication.

2. 2

   Set the two bases.

   $A = 2x+1, \qquad B = 2x-1$

   Both are linear in $x$ with the same leading coefficient, which is what makes one of the factors collapse to a constant.

3. 3

   Compute A − B. Distribute the minus sign over the whole second bracket:

   $(2x+1) - (2x-1) = 2x + 1 - 2x + 1 = 2$

   The $2x$ terms cancel; forgetting to flip the sign of $-1$ would give $0$ instead of $2$ and wipe out the answer.

4. 4

   Compute A + B and multiply.

   $(2x+1) + (2x-1) = 4x$

   So

   $(2x+1)^2 - (2x-1)^2 = 2 \cdot 4x = 8x$

5. 5

   Check against direct expansion. Expanding gives

   $(4x^2 + 4x + 1) - (4x^2 - 4x + 1) = 8x$

   The $4x^2$ and the $+1$ cancel, leaving only the cross terms — which is the structural reason the result is linear. At $x = 5$ both forms give $121 - 81 = 40 = 8 \cdot 5$.