Simplify the following:

$\begin{matrix} \text{(i)} & (5 + \sqrt{7})(2 + \sqrt{5}) \\[1.5em] \text{(ii)} & (5 + \sqrt{5})(5 - \sqrt{5}) \\[1.5em] \text{(iii)} & (\sqrt{5} + \sqrt{2})^2 \\[1.5em] \text{(iv)} & (\sqrt{3} - \sqrt{7})^2 \\[1.5em] \text{(v)} & (\sqrt{2} + \sqrt{3})(\sqrt{5} + \sqrt{7}) \\[1.5em] \text{(vi)} & (4 + \sqrt{5})(\sqrt{3} - \sqrt{7}) \\[1.5em] \end{matrix}$

$\text{(i) } (5 + \sqrt{7})(2 + \sqrt{5}) \\[1.5em] = 10 + 5\sqrt{5} + 2\sqrt{7} + \sqrt{7}\sqrt{5} \\[1.5em] = \bold{10 + 5\sqrt{5} + 2\sqrt{7} + \sqrt{35}} \\[1.5em]$

$\text{(ii) } (5 + \sqrt{5})(5 - \sqrt{5}) \\[1.5em] \text{Using identity} : (a + b)(a - b) = a^2 - b^2 \\[1.5em] = 5^2 - (\sqrt{5})^2 \\[1.5em] = \bold{25 - 5 = 20} \\[1.5em]$

$\text{(iii) } (\sqrt{5} + \sqrt{2})^2 \\[1.5em] \text{Using identity} : (a + b)^2 = a^2 + 2ab + b^2 \\[1.5em] = (\sqrt{5} + \sqrt{2})^2 = {(\sqrt{5})}^2 + 2 × \sqrt{5} ×\sqrt{2} + {(\sqrt{2})}^2 \\[1.5em] = 5 + 2\sqrt{10} + 2 \\[1.5em] = \bold{7 + 2\sqrt{10}} \\[1.5em]$

$\text{(iv) } (\sqrt{3} - \sqrt{7})^2 \\[1.5em] \text{Using identity} : (a - b)^2 = a^2 - 2ab + b^2 \\[1.5em] = (\sqrt{3} - \sqrt{7})^2 = {(\sqrt{3})}^2 - 2 × \sqrt{3} ×\sqrt{7} + {(\sqrt{7})}^2 \\[1.5em] = 3 - 2\sqrt{21} + 7 \\[1.5em] = \bold{10 - 2\sqrt{21}} \\[1.5em]$

$\text{(v) } (\sqrt{2} + \sqrt{3})(\sqrt{5} + \sqrt{7}) \\[1.5em] = \sqrt{2} × \sqrt{5} + \sqrt{2} × \sqrt{7} + \sqrt{3} × \sqrt{5} + \sqrt{3} × \sqrt{7} \\[1.5em] = \bold{\sqrt{10} + \sqrt{14} + \sqrt{15} + \sqrt{21}} \\[1.5em]$

$\text{(vi) } (4 + \sqrt{5})(\sqrt{3} - \sqrt{7}) \\[1.5em] = 4\sqrt{3} - 4\sqrt{7} + \sqrt{5} × \sqrt{3} - \sqrt{5} × \sqrt{7} \\[1.5em] = \bold{4\sqrt{3} - 4\sqrt{7} + \sqrt{15} - \sqrt{35}} \\[1.5em]$