Answer: \(a=-\frac14,\ b=\frac14\).

Since

\((\sqrt5+2)(\sqrt5-2)=5-4=1,\)

we have

\(\frac1{\sqrt5+2}=\sqrt5-2\)

and

\(\frac1{\sqrt5-2}=\sqrt5+2.\)

Therefore

So

\(a(\sqrt5-2)+b(\sqrt5+2) =(a+b)\sqrt5+2b-2a.\)

This must be equal to \(1\), which is rational. Hence the coefficient of \(\sqrt5\) must be zero:

\(a+b=0.\)

The rational part must be \(1\), so

\(2b-2a=1.\)

From \(a+b=0\), \(b=-a\). Substitute this into \(2b-2a=1\):

\(2(-a)-2a=1.\)

Thus

\(-4a=1,\)

so

\(a=-\frac14.\)

Then

\(b=\frac14.\)