Expand and simplify (1 + root 2)(root 2 - 4).

Step-by-step solution

1. 1

   Multiply out all four products. Using FOIL on the two binomials:

   $1\cdot\sqrt{2}=\sqrt{2},\qquad 1\cdot(-4)=-4,$
   $\sqrt{2}\cdot\sqrt{2}=2,\qquad \sqrt{2}\cdot(-4)=-4\sqrt{2}.$

2. 2

   Note the key simplification. $\sqrt{2}\cdot\sqrt{2}=2$ — a square root times itself returns the number underneath. Leaving it as $\sqrt{4}$ or, worse, as $2\sqrt2$ is the usual slip.

3. 3

   Add the four pieces.

   $\sqrt{2}-4+2-4\sqrt{2}.$

4. 4

   Group rational parts with rational parts and surds with surds.

   $\underbrace{(-4+2)}_{\text{rational}}+\underbrace{\left(\sqrt{2}-4\sqrt{2}\right)}_{\text{surd}}=-2-3\sqrt{2}.$

   The surd terms combine like $1x-4x=-3x$ because they share the same radicand.

5. 5

   Check numerically. $\sqrt2=1.414214$, so the original is $(2.414214)(-2.585786)=-6.242641$, and $-2-3(1.414214)=-6.242641$. They agree to every digit.