Answer
$8-2\sqrt{10}+2\sqrt{5}-2\sqrt{2}$
Work Step by Step
Using $(a+b+c)=a^2+b^2+c^2+2ab+2ac+2bc$ (or the square of a multinomial), the given expression, $
(\sqrt{5}-\sqrt{2}+1)^2
,$ is equivalent to
\begin{array}{l}\require{cancel}
(\sqrt{5})^2+(-\sqrt{2})^2+(1)^2+2(\sqrt{5})(-\sqrt{2})+2(\sqrt{5})(1)+2(-\sqrt{2})(1)
\\\\=
5+2+1-2\sqrt{5(2)}+2\sqrt{5}-2\sqrt{2}
\\\\=
8-2\sqrt{10}+2\sqrt{5}-2\sqrt{2}
.\end{array}