Simplify \(\sqrt{8}\) to \(2\sqrt{2}\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{(2\sqrt{2}-\sqrt{20})}^{2}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}-\sqrt{20})}\]
Simplify \(\sqrt{20}\) to \(2\sqrt{5}\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{(2\sqrt{2}-2\sqrt{5})}^{2}}{(\sqrt{2}+\sqrt{5})(2\sqrt{2}-2\sqrt{5})}\]
Factor out the common term \(2\).
\[\frac{{(\sqrt{2}+\sqrt{5})}^{2}+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Expand.
\[\frac{{\sqrt{2}}^{2}+2\sqrt{2}\sqrt{5}+{\sqrt{5}}^{2}+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{2+2\sqrt{2}\sqrt{5}+{\sqrt{5}}^{2}+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{2+2\sqrt{2}\sqrt{5}+5+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Simplify \(2\sqrt{2}\sqrt{5}\) to \(2\sqrt{2\times 5}\).
\[\frac{2+2\sqrt{2\times 5}+5+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Simplify \(2\times 5\) to \(10\).
\[\frac{2+2\sqrt{10}+5+{(2(\sqrt{2}-\sqrt{5}))}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Collect like terms.
\[\frac{(2+5)+2\sqrt{10}+4{(\sqrt{2}-\sqrt{5})}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Simplify \((2+5)+2\sqrt{10}+4{(\sqrt{2}-\sqrt{5})}^{2}\) to \(7+2\sqrt{10}+4{(\sqrt{2}-\sqrt{5})}^{2}\).
\[\frac{7+2\sqrt{10}+4{(\sqrt{2}-\sqrt{5})}^{2}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Expand.
\[\frac{7+2\sqrt{10}+8-8\sqrt{10}+20}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Collect like terms.
\[\frac{(7+8+20)+(2\sqrt{10}-8\sqrt{10})}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Simplify \((7+8+20)+(2\sqrt{10}-8\sqrt{10})\) to \(35-6\sqrt{10}\).
\[\frac{35-6\sqrt{10}}{(\sqrt{2}+\sqrt{5})\times 2(\sqrt{2}-\sqrt{5})}\]
Regroup terms.
\[\frac{35-6\sqrt{10}}{2(\sqrt{2}+\sqrt{5})(\sqrt{2}-\sqrt{5})}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{35-6\sqrt{10}}{2({\sqrt{2}}^{2}-{\sqrt{5}}^{2})}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{35-6\sqrt{10}}{2(2-{\sqrt{5}}^{2})}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{35-6\sqrt{10}}{2(2-5)}\]
Simplify \(2-5\) to \(-3\).
\[\frac{35-6\sqrt{10}}{2\times -3}\]
Simplify \(2\times -3\) to \(-6\).
\[\frac{35-6\sqrt{10}}{-6}\]
Move the negative sign to the left.
\[-\frac{35-6\sqrt{10}}{6}\]
Decimal Form: -2.671056
-(35-6*sqrt(10))/6
