To find the simplified form of the quotient
[tex]\[
\frac{2 - \sqrt{8}}{4 + \sqrt{12}},
\][/tex]
we will proceed through several steps.
### Step 1: Simplify the Radical Terms
First, let's simplify the radical terms:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}, \quad \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}.
\][/tex]
Substituting these into the original expression, we get:
[tex]\[
\frac{2 - 2\sqrt{2}}{4 + 2\sqrt{3}}.
\][/tex]
### Step 2: Rationalize the Denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(4 - 2\sqrt{3}\)[/tex]:
[tex]\[
\frac{(2 - 2\sqrt{2})(4 - 2\sqrt{3})}{(4 + 2\sqrt{3})(4 - 2\sqrt{3})}.
\][/tex]
### Step 3: Multiply the Numerator
Let's expand the numerator:
[tex]\[
(2 - 2\sqrt{2})(4 - 2\sqrt{3}) = 2 \cdot 4 + 2 \cdot (-2\sqrt{3}) - 2\sqrt{2} \cdot 4 - 2\sqrt{2} \cdot (-2\sqrt{3}).
\][/tex]
Simplifying each term:
[tex]\[
= 8 - 4\sqrt{3} - 8\sqrt{2} + 4\sqrt{6}.
\][/tex]
So, the simplified numerator is:
[tex]\[
8 - 8\sqrt{2} - 4\sqrt{3} + 4\sqrt{6}.
\][/tex]
### Step 4: Multiply the Denominator
Now, we expand the denominator:
[tex]\[
(4 + 2\sqrt{3})(4 - 2\sqrt{3}) = 4^2 - (2\sqrt{3})^2.
\][/tex]
Calculating the terms:
[tex]\[
= 16 - 4 \cdot 3 = 16 - 12 = 4.
\][/tex]
### Step 5: Simplify the Expression
Combining the results from steps 3 and 4:
[tex]\[
\frac{8 - 8\sqrt{2} - 4\sqrt{3} + 4\sqrt{6}}{4}.
\][/tex]
We can simplify this by dividing each term in the numerator by 4:
[tex]\[
\frac{8}{4} - \frac{8\sqrt{2}}{4} - \frac{4\sqrt{3}}{4} + \frac{4\sqrt{6}}{4}
= 2 - 2\sqrt{2} - \sqrt{3} + \sqrt{6}.
\][/tex]
So, the simplified quotient is:
[tex]\[
2 - 2\sqrt{2} - \sqrt{3} + \sqrt{6}.
\][/tex]
This matches the choice:
[tex]\[
\boxed{2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6}}.
\][/tex]
