Let $(x-2)$ is a factor of the given expression. $x-2\;=0$ $x\;=2$Given: $2 x^{3}+a x^{2}+b x-14=0$ $2(2)^{3}+a(2)^{2}+b(2)-14\;=0$ $16+4 a+2 b-14\;=0$ $4 a+2 b+2\;=0$ $4 a+2 b\;=-2$ $2 a+b\;=-1 \;\; \ldots \;\text{(i)}\;$and when given expression is divided by $(x-3)$ . $x-3\;=0$ $x \;\; =3$ $\therefore\;\; 2 x^{3}+a x^{2}+b x-14\;=52$ $2(3)^{3}+a(3)^{2}+b(3)-66\;=0$ $54+9 a+3 b-66\;=0$ $9 a+3 b\;=12$ $3 a+b=4$ ............(ii)Solving equation (i) and (ii), $\;2 a+b=-1$ $\;3 a+b=4$ $(-)(-) \;\; (-)$ $--------$ $-a=-5 \;\; \Rightarrow a=5$from (ii), $3 \times 5+b\;=4$ $b\;=4-15=-11$ $\therefore\;\; a=5 \;\text{and}\; b=-11$