Answer
$$ - 144 - 8\sqrt {10} $$
Work Step by Step
$$\eqalign{
& \left| {\matrix{
{\sqrt 2 } & 4 & 0 \cr
1 & { - \sqrt 5 } & 7 \cr
{ - 5} & {\sqrt 5 } & 1 \cr
} } \right| \cr
& {\rm{Calculating\, the\, determinant\, by \,expanding\, the\, third\, column}} \cr
& \left| {\matrix{
{\sqrt 2 } & 4 & 0 \cr
1 & { - \sqrt 5 } & 7 \cr
{ - 5} & {\sqrt 5 } & 1 \cr
} } \right| = 0\left| {\matrix{
1 & { - \sqrt 5 } \cr
{ - 5} & {\sqrt 5 } \cr
} } \right| - 7\left| {\matrix{
{\sqrt 2 } & 4 \cr
{ - 5} & {\sqrt 5 } \cr
} } \right| + 1\left| {\matrix{
{\sqrt 2 } & 4 \cr
1 & { - \sqrt 5 } \cr
} } \right| \cr
& {\rm{Solving}} \cr
& \left| {\matrix{
{\sqrt 2 } & 4 & 0 \cr
1 & { - \sqrt 5 } & 7 \cr
{ - 5} & {\sqrt 5 } & 1 \cr
} } \right| = - 7\left( {\sqrt {10} + 20} \right) + \left( { - \sqrt {10} - 4} \right) \cr
& {\rm{Simplifying}} \cr
& \left| {\matrix{
{\sqrt 2 } & 4 & 0 \cr
1 & { - \sqrt 5 } & 7 \cr
{ - 5} & {\sqrt 5 } & 1 \cr
} } \right| = - 7\sqrt {10} - 140 - \sqrt {10} - 4 \cr
& \left| {\matrix{
{\sqrt 2 } & 4 & 0 \cr
1 & { - \sqrt 5 } & 7 \cr
{ - 5} & {\sqrt 5 } & 1 \cr
} } \right| = - 144 - 8\sqrt {10} \cr} $$
