Of course! Let's expand and simplify the expression step-by-step. We aim to expand the expression [tex]\((\sqrt{2} x + 2 \sqrt{2} y + z) \left(2 x^2 + 8 y^2 + z^2 - 4 x y - 2 \sqrt{2} y z - \sqrt{2} x z\right)\)[/tex].
### Step 1: Distribute [tex]\(\sqrt{2} x\)[/tex] across each term within the parentheses:
First, distribute [tex]\(\sqrt{2} x\)[/tex]:
[tex]\[ \sqrt{2} x \cdot 2 x^2 = 2\sqrt{2} x^3 \][/tex]
[tex]\[ \sqrt{2} x \cdot 8 y^2 = 8\sqrt{2} x y^2 \][/tex]
[tex]\[ \sqrt{2} x \cdot z^2 = \sqrt{2} x z^2 \][/tex]
[tex]\[ \sqrt{2} x \cdot (-4 x y) = -4\sqrt{2} x^2 y \][/tex]
[tex]\[ \sqrt{2} x \cdot (-2 \sqrt{2} y z) = -4 x y z \][/tex] (since [tex]\(\sqrt{2} \cdot \sqrt{2} = 2\)[/tex])
[tex]\[ \sqrt{2} x \cdot (-\sqrt{2} x z) = -2 x^2 z \][/tex]
### Step 2: Distribute [tex]\(2\sqrt{2} y\)[/tex] across each term within the parentheses:
Next, distribute [tex]\(2 \sqrt{2} y\)[/tex]:
[tex]\[ 2\sqrt{2} y \cdot 2 x^2 = 4\sqrt{2} x^2 y \][/tex]
[tex]\[ 2\sqrt{2} y \cdot 8 y^2 = 16\sqrt{2} y^3 \][/tex]
[tex]\[ 2\sqrt{2} y \cdot z^2 = 2\sqrt{2} y z^2 \][/tex]
[tex]\[ 2\sqrt{2} y \cdot (-4 x y) = -8\sqrt{2} x y^2 \][/tex]
[tex]\[ 2\sqrt{2} y \cdot (-2 \sqrt{2} y z) = -8 y^2 z \][/tex]
[tex]\[ 2\sqrt{2} y \cdot (-\sqrt{2} x z) = -4 x y z \][/tex]
### Step 3: Distribute [tex]\(z\)[/tex] across each term within the parentheses:
Finally, distribute [tex]\(z\)[/tex]:
[tex]\[ z \cdot 2 x^2 = 2 x^2 z \][/tex]
[tex]\[ z \cdot 8 y^2 = 8 y^2 z \][/tex]
[tex]\[ z \cdot z^2 = z^3 \][/tex]
[tex]\[ z \cdot (-4 x y) = -4 x y z \][/tex]
[tex]\[ z \cdot (-2 \sqrt{2} y z) = -2 \sqrt{2} y z^2 \][/tex]
[tex]\[ z \cdot (-\sqrt{2} x z) = -\sqrt{2} x z^2 \][/tex]
### Step 4: Combine like terms:
Now we combine all the terms we obtained. Group similar terms to simplify:
[tex]\[ 2\sqrt{2} x^3 + 4\sqrt{2} x^2 y - 4\sqrt{2} x^2 y + 8\sqrt{2} x y^2 - 8\sqrt{2} x y^2 - 2 x^2 z + 2 x^2 z \][/tex]
[tex]\[ -4 x y z - 4 x y z - 8 y^2 z + 16\sqrt{2} y^3 + x z^2 + \sqrt{2} x z^2 - \sqrt{2} x z^2 + z^3 - 2 \sqrt{2} y z^2 + 2\sqrt{2} y z^2 + 8 y^2 z \][/tex]
Notice that almost all of these terms cancel out, except for these:
[tex]\[ 2\sqrt{2} x^3 + 16\sqrt{2} y^3 + z^3 - 12 x y z \][/tex]
So the resultant expanded form of the expression [tex]\((\sqrt{2} x + 2 \sqrt{2} y + z) \left(2 x^2 + 8 y^2 + z^2 - 4 x y - 2 \sqrt{2} y z - \sqrt{2} x z\right)\)[/tex] is:
[tex]\[ \boxed{2\sqrt{2} x^3 + 16\sqrt{2} y^3 + z^3 - 12 x y z}\][/tex]
