Expressing in rationals I have this question to express it in a specific form

Express $\dfrac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}$ in the form $a \sqrt{2} + b \sqrt{3} + c \sqrt{7} + d \sqrt{42}$, for some rationals $a,b,c$ and $d$.

So the solution that I have and am trying to understand is this

So what I don't understand is that why in the world is there a 2sqrt6 here where did it come from?

Just for the record, elaborating on Berci's suggestion, you could also try solving 
$$(a \sqrt{2} + b \sqrt{3} + c \sqrt{7} + d \sqrt{42})(\sqrt{2} + \sqrt{3} + \sqrt{7}) = 1.$$
Writing out the product, you get
$$(2a + 3b + 7c) + \sqrt{6}(a + b + 7d) + \sqrt{14}(a + c + 3d) + \sqrt{21}(b + c + 2d) = 1.$$
With all coefficients rational, this implies that the following matrix equation has to be satisfied:
$$\begin{pmatrix} 2 & 3 & 7 & 0 \\ 1 & 1 & 0 & 7 \\ 1 & 0 & 1 & 3 \\ 0 & 1 & 1 & 2 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}.$$
There are several ways to solve such systems, all of them leading to the solution:
$$\begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = \frac{1}{10} \begin{pmatrix} 4 \\ 3 \\ -1 \\ -1 \end{pmatrix}.$$
This does not involve any tricks with getting rid of square roots, although one may argue this way to get to the result takes more time.
