Explanation

(d)
$x = \left( \sqrt{3} + 1 \right)^{5} = \left( \sqrt{3} \right)^{5} + \ ^{5}C_{1}\left( \sqrt{3} \right)^{4} + {\ ^{5}C_{2}\left( \sqrt{3} \right)}^{3}$
$+ \ ^{5}C_{3}\left( \sqrt{3} \right)^{2} + \ ^{5}C_{4}(\sqrt{3)} + \ ^{5}C_{5}$
$= 9\sqrt{3} + 45 + 30\sqrt{3} + 30 + 5\sqrt{3} + 1$
$= 76 + 44\sqrt{3}$
$\therefore\ \lbrack x\rbrack = \left\lbrack \left( \sqrt{3} + 1 \right)^{5} \right\rbrack = \left\lbrack 76 + 44\sqrt{3} \right\rbrack$
$= \lbrack 76\rbrack + \lbrack 44 \times 1.732\rbrack$
$= 76 + \lbrack 76.2\rbrack$
$= 76 + 76 = 152$