In the expansion of $\left(1+a x+b x^{2}\right)(1-2 x)^{18},$

Coefficient of $x^{3}$ in $\left(1+a x+b x^{2}\right)(1-2 x)^{18}$

$=$ Coefficient of $x^{3}$ in $(1-2 x)^{18}$

${+\text { Coefficient of } x^{2} \text { in a }(1-2 x)^{\text {18 }}}$

${\text { + Coefficient of } x \text { in } b(1-2 x)^{\text {18 }}}$

$=-^{18} \mathrm{C}_{3} \cdot 2^{3}+\mathrm{a}^{18} \mathrm{C}_{2} \cdot 2^{2}-\mathrm{b}^{18} \mathrm{C}_{1} \cdot 2$

$\because-^{18} C_{3} \cdot 2^{3}+a^{18} C_{2} \cdot 2^{2}-b^{18} C_{1} \cdot 2=0$

$\Rightarrow \frac{18 \times 17 \times 16}{3 \times 2} \cdot 8+a \cdot \frac{18 \times 17}{2} 2^{2}-b \cdot 18 \cdot 2=0$

$\Rightarrow 17 a-b=\frac{34 \times 16}{3}……….(i)$

Similarly, coefficient of $x^{4}$

$^{18} \mathrm{C}_{4} \cdot 2^{4}-\mathrm{a} \cdot^{18} \mathrm{C}_{3} 2^{3}+b \cdot^{18} \mathrm{C}_{2} \cdot 2^{2}=0$

$32 a-32 b=240……….(ii)$

On solving Eqs. $(i)$ and $(ii)$, we get

$a=16 \text { and } b=\frac{272}{3}$