To solve the quotient
[tex]\[ \frac{2 - \sqrt{8}}{4 + \sqrt{12}} \][/tex]
we need to simplify both the numerator and the denominator.
### Step 1: Simplify the numerator and denominator
First, let's look at the numerator:
[tex]\[ 2 - \sqrt{8} \][/tex]
We know that [tex]\( \sqrt{8} = 2\sqrt{2} \)[/tex], so we can rewrite the numerator as:
[tex]\[ 2 - 2\sqrt{2} \][/tex]
Next, the denominator:
[tex]\[ 4 + \sqrt{12} \][/tex]
We know that [tex]\( \sqrt{12} = 2\sqrt{3} \)[/tex], so we can rewrite the denominator as:
[tex]\[ 4 + 2\sqrt{3} \][/tex]
### Step 2: Rationalize the denominator
To get rid of the surd in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\( 4 + 2\sqrt{3} \)[/tex] is [tex]\( 4 - 2\sqrt{3} \)[/tex].
So we multiply:
[tex]\[ \frac{(2 - 2\sqrt{2})(4 - 2\sqrt{3})}{(4 + 2\sqrt{3})(4 - 2\sqrt{3})} \][/tex]
### Step 3: Expand the numerator and denominator
Let's expand the numerator [tex]\((2 - 2\sqrt{2})(4 - 2\sqrt{3})\)[/tex]:
[tex]\[
\begin{align*}
(2 - 2\sqrt{2})(4 - 2\sqrt{3}) &= 2 \cdot 4 - 2 \cdot 2\sqrt{3} - 2\sqrt{2} \cdot 4 + 2\sqrt{2} \cdot 2\sqrt{3} \\
&= 8 - 4\sqrt{3} - 8\sqrt{2} + 4\sqrt{6}
\end{align*}
\][/tex]
Now, expand the denominator [tex]\((4 + 2\sqrt{3})(4 - 2\sqrt{3})\)[/tex]:
[tex]\[
\begin{align*}
(4 + 2\sqrt{3})(4 - 2\sqrt{3}) &= 4 \cdot 4 - (2\sqrt{3})^2 \\
&= 16 - 12 \\
&= 4
\end{align*}
\][/tex]
### Step 4: Simplify the final expression
Putting it all together:
[tex]\[
\frac{8 - 4\sqrt{3} - 8\sqrt{2} + 4\sqrt{6}}{4}
\][/tex]
We can simplify this by dividing each term in the numerator by 4:
[tex]\[
8/4 - 4\sqrt{3}/4 - 8\sqrt{2}/4 + 4\sqrt{6}/4 = 2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6}
\][/tex]
### Result
Thus, the final simplified form of the quotient is:
[tex]\[ 2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6} \][/tex]
This is the answer to the question.
