Cardano is the way to go

Relevant wiki: Cardano's Method

Consider Cardano's method.

$a=1, \quad b=-3, \quad c=-2, \quad d=4-6\sqrt{2}.$

Compute $Q$ and $R:$

$\begin{aligned} Q &= \frac {3 a c - b^2} {9 a^2} \\ \\ &=-\frac{5}{3} \\ \\ \\ R &= \frac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3} \\ \\ &= 3\sqrt{2}. \end{aligned}$

Compute $S$ and $T:$

$\begin{aligned} S &= \sqrt [3] {R + \sqrt{Q^3 + R^2}} \\ \\ &= \sqrt[3]{3\sqrt{2}+\frac{19\sqrt{3}}{9}} \\ \\ \\ T &= \sqrt [3] {R - \sqrt{Q^3 + R^2}} \\ \\ &= \sqrt[3]{3\sqrt{2}-\frac{19\sqrt{3}}{9}} \end{aligned}$

The real root will be:

$\begin{aligned} x_1 &= S + T - \frac b {3 a} \\ \\ &= \sqrt[3]{3\sqrt{2}+\frac{19\sqrt{3}}{9}} + \sqrt[3]{3\sqrt{2}-\frac{19\sqrt{3}}{9}} +1 \\ \\ &\approx \boxed{3.828} \end{aligned}$

Incidentally, this value is equal to $2\sqrt{2}+1.$ This shows a weakness of Cardano's method: the result tends to contain a sum of cubic roots which isn't expressed in the simplest possible way.