Question 100
Given the equation
$$
2 + x\sqrt{9 + 6\sqrt{2}} = x\sqrt{5 - 2\sqrt{6}} + \sqrt{6} - 2\sqrt{3} + \sqrt{2}.
$$
a) Write the root of the equation in the form $m - \sqrt{n}$, where $m$ and $n$ are natural numbers.
b) Factor the expression $a^3 - 3a^2 - 5a + 7$ into two non-constant factors with integer coefficients and calculate the value of this expression if $a$ is the root found in a).
$$
2 + x\sqrt{9 + 6\sqrt{2}} = x\sqrt{5 - 2\sqrt{6}} + \sqrt{6} - 2\sqrt{3} + \sqrt{2}.
$$
a) Write the root of the equation in the form $m - \sqrt{n}$, where $m$ and $n$ are natural numbers.
b) Factor the expression $a^3 - 3a^2 - 5a + 7$ into two non-constant factors with integer coefficients and calculate the value of this expression if $a$ is the root found in a).