Answer
$4-2\sqrt{10}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
[(\sqrt{5}-\sqrt{2})-\sqrt{3}][(\sqrt{5}-\sqrt{2})+\sqrt{3}]
,$ use the special products on multiplying the sum and difference of like terms and squaring binomials. Then, combine like terms.
$\bf{\text{Solution Details:}}$
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
(\sqrt{5}-\sqrt{2})^2-(\sqrt{3})^2
\\\\=
(\sqrt{5}-\sqrt{2})^2-3
.\end{array}
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
[(\sqrt{5})^2-2(\sqrt{5})(\sqrt{2})+(\sqrt{2})^2-3
\\\\=
5-2(\sqrt{5})(\sqrt{2})+2-3
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
5-2\sqrt{5(2)}+2-3
\\\\=
5-2\sqrt{10}+2-3
.\end{array}
Combining like terms, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(5+2-3)-2\sqrt{10}
\\\\=
4-2\sqrt{10}
.\end{array}
