Answer: \(a=3\), \(b=5\), \(c=3051\).

Use the binomial theorem and identify the required term carefully from the general term. Keep the powers of the variable explicit so that the required coefficient or constant term can be read off clearly.

The first terms of \((ax-2)^4\) are

\(a^4x^4-8a^3x^3+24a^2x^2+\cdots.\)

The first terms of \(\left(1+\frac{b}{x}\right)^3\) are

\(1+\frac{3b}{x}+\frac{3b^2}{x^2}+\cdots.\)

The coefficient of \(x^4\) is \(a^4\), so

\(a^4=81.\)

Since \(a\) is positive, \(a=3\).

The coefficient of \(x^3\) is

\(-8a^3+3a^4b.\)

Using \(a=3\),

\(-8(27)+3(81)b=999.\)

So

\(-216+243b=999,\)

and hence \(b=5\).

The coefficient of \(x^2\) is

\(24a^2+(-8a^3)(3b)+a^4(3b^2).\)

Substitute \(a=3\) and \(b=5\):

\(24(9)-8(27)(15)+81(75)=216-3240+6075=3051.\)

Thus \(c=3051\).

This gives the required answer: \(a=3\), \(b=5\), \(c=3051\).