Simplify the product (2x - sqrt(2x) - 1)(2x + sqrt(2x) - 1).

Step-by-step solution

1. 1

   Look for the conjugate pattern rather than multiplying out nine terms. The two factors are identical except for the sign in front of $\sqrt{2x}$. That is the signature of $(a-b)(a+b) = a^2 - b^2$, provided the terms are grouped correctly.

2. 2

   Choose the grouping. Put everything that does not change sign into $a$ and the radical into $b$:

   $a = 2x - 1, \qquad b = \sqrt{2x}$

   Then the product is exactly $\left(a - b\right)\left(a + b\right)$.

3. 3

   Apply the identity.

   $\left(a-b\right)\left(a+b\right) = a^2 - b^2 = (2x-1)^2 - \left(\sqrt{2x}\right)^2$

   The radical disappears in one step — this is why the grouping is worth spotting.

4. 4

   Expand each square.

   $(2x-1)^2 = 4x^2 - 4x + 1, \qquad \left(\sqrt{2x}\right)^2 = 2x$

5. 5

   Subtract and collect.

   $4x^2 - 4x + 1 - 2x = 4x^2 - 6x + 1$

6. 6

   Check numerically at x = 2. The factors are $4 - 2 - 1 = 1$ and $4 + 2 - 1 = 5$, product $5$. The simplified form gives $4(4) - 6(2) + 1 = 16 - 12 + 1 = 5$. They match. (Note the original expression needs $x \ge 0$ for the radical, while the polynomial form is defined for all $x$.)