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[tex]\frac{\sqrt{2}}{\sqrt{2} - 1} + \frac{4}{\sqrt{2} + 1} = \frac{(\sqrt{2})(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} + \frac{(4)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} [/tex]
[tex]= \frac{(\sqrt{2})(\sqrt{2} + 1) + (4)(\sqrt{2} - 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} [/tex]
[tex]= \frac{\sqrt{4} + \sqrt{2} + 4 \sqrt{2} - 4}{\sqrt{4} + \sqrt{2} - \sqrt{2} - 1} [/tex]
[tex]= \frac{2 + (1 + 4) \sqrt{2} - 4}{\sqrt{4} - 1} = \frac{2 - 4 + 5 \sqrt{2}}{2 - 1} = \frac{ - 2 + 5 \sqrt{2}}{1} = 5 \sqrt{2} - 2[/tex]
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