Let $\lambda \ne 0$  be a real number.  Let $\alpha ,\beta$  be the roots of the equation  $14x^2-31x+3\lambda =0$  and $\alpha ,\gamma$   be the roots of the equation  $35x^2-53x+4\lambda =0$ . Then $\frac{3\alpha }{\beta }$   and $\frac{4\alpha }{\gamma }$   are the roots of the equation.

Eliminating $\lambda$  , we get  $\alpha =\frac{5}{7}$
 Here   $\alpha +\beta =\frac{31}{14}$  $\Rightarrow$  $\beta =\frac{31}{14}-\frac{5}{7}=\frac{3}{2}$  and  $\alpha +\gamma =\frac{53}{35}$$\Rightarrow$  $\gamma =\frac{4}{5}$
Required equation is $x^2-(\frac{3\alpha }{\beta }+\frac{4\alpha }{\gamma })x +(\frac{3\alpha }{\beta }.\frac{4\alpha }{\gamma })=0$

  $$\begin{gathered} x^2-(\frac{10}{7}+\frac{25}{7})x+\frac{250}{49}=0 \\ \Rightarrow 49x^2-245x+250=0 \end{gathered}$$