$\text{ If }\alpha \text{ and }\beta \text{ are the roots of the equation, }7x^2-3x-2=0,\text{ then the value of }$$\frac{\alpha }{1-{\alpha }^2}+\frac{\beta }{1-{\beta }^2}isequalto:$

$\text{ Given }\alpha \text{ and }\beta \text{ are the roots of the equation, }7x^2-3x-2=0\text{ , hence }\alpha +\beta =\frac{3}{7},\alpha \beta =-\frac{2}{7}$

Consider 

$\begin{array}{c}{\alpha }^2+{\beta }^2={\left(\alpha +\beta \right)}^2-2\alpha \beta =\frac{9}{49}+\frac{4}{7}==\frac{37}{49}\\ \end{array}$

And the required expression is 

$\begin{array}{c}\frac{\alpha }{1-{\alpha }^2}+\frac{\beta }{1-{\beta }^2}=\frac{\alpha +\beta -\alpha \beta \left(\alpha +\beta \right)}{1-\left({\alpha }^2+{\beta }^2\right)+{\left(\alpha \beta \right)}^2}\\ =\frac{\frac{3}{7}+\frac{6}{49}}{1-\frac{37}{49}+\frac{4}{49}}\\ =\frac{27}{16}\end{array}$