Answer
$$2 - 4\sqrt {21} $$
Work Step by Step
$$\eqalign{
& \left| {\matrix{
{\sqrt 3 } & 1 & 0 \cr
{\sqrt 7 } & 4 & { - 1} \cr
5 & 0 & { - \sqrt 7 } \cr
} } \right| \cr
& {\rm{Calculating\, the\, determinant \,by \,expanding\, the\, third\, column}} \cr
& \left| {\matrix{
{\sqrt 3 } & 1 & 0 \cr
{\sqrt 7 } & 4 & { - 1} \cr
5 & 0 & { - \sqrt 7 } \cr
} } \right| = 0\left| {\matrix{
{\sqrt 7 } & 4 \cr
5 & 0 \cr
} } \right| - \left( { - 1} \right)\left| {\matrix{
{\sqrt 3 } & 1 \cr
5 & 0 \cr
} } \right| - \sqrt 7 \left| {\matrix{
{\sqrt 3 } & 1 \cr
{\sqrt 7 } & 4 \cr
} } \right| \cr
& {\rm{Solving}} \cr
& \left| {\matrix{
{\sqrt 3 } & 1 & 0 \cr
{\sqrt 7 } & 4 & { - 1} \cr
5 & 0 & { - \sqrt 7 } \cr
} } \right| = 0 + \left( {0 - 5} \right) - \sqrt 7 \left( {4\sqrt 3 - \sqrt 7 } \right) \cr
& {\rm{Simplifying}} \cr
& \left| {\matrix{
{\sqrt 3 } & 1 & 0 \cr
{\sqrt 7 } & 4 & { - 1} \cr
5 & 0 & { - \sqrt 7 } \cr
} } \right| = - 5 - 4\sqrt {21} + 7 \cr
& \left| {\matrix{
{\sqrt 3 } & 1 & 0 \cr
{\sqrt 7 } & 4 & { - 1} \cr
5 & 0 & { - \sqrt 7 } \cr
} } \right| = 2 - 4\sqrt {21} \cr} $$
