\[10+ 4 \sqrt{6}\]
let \[(a+b \sqrt{6})^2=10+4 \sqrt{6}\]
.Expand the brackets
\[a^2+2ab \sqrt{6} +6b^2=10+4 \sqrt{6}\]
.Not equate the integers and the square roots to get simultaneous equations.
\[a^2+6b^2=10\]
\[2ab=4 \rightarrow ab=2\]
Since we are looking for integer solution, the second equation gives
\[a= \pm 1, \: b= \pm 2\]
or \[a= \pm 2, \: b= \pm 1\]
&.Only
\[a= \pm 2, \: b= \pm 1\]
fits the first of the simultaneous equations, so the solutions are \[a=-2, \: b=-1\]
or \[a=2, \: b=1\]
.