Answer: (a) \(a=243\), \(b=-45\), \(c=\frac{10}{3}\); (b) coefficient \(=\frac{469}{3}\).

Answer: (a) \(a=243\), \(b=-45\), \(c=\frac{10}{3}\); (b) coefficient \(=\frac{469}{3}\).

(a) Use the binomial expansion. The first term is

\((3x^2)^5=243x^{10}.\)

The next term is

\(\binom51(3x^2)^4\left(-\frac{1}{9x}\right)=5\cdot81x^8\left(-\frac{1}{9x}\right)=-45x^7.\)

The third term is

So

\(a=243,\qquad b=-45,\qquad c=\frac{10}{3}.\)

(b) Also,

To obtain an \(x^4\) term, use

Therefore the coefficient of \(x^4\) is

\(\frac{10}{3}+2(-45)+243=\frac{469}{3}.\)