<s>[INST] Let $\alpha $, $\beta $ are the roots of $3x^2+x+5=0$ Let $\alpha $,  $\beta $ are the roots of $3x^2+x+5=0$ then find a quadratic equation with roots as $\dfrac {\alpha +1}{\alpha -3}$ and $\dfrac {\beta +1}{\beta -3}$.

I got this solution but didn't understand what procedure is used,  what concepts are used and how's it working? 
 [/INST] For a different route, try this.
Let $a=\dfrac {\alpha +1}{\alpha -3}$ and $b=\dfrac {\beta +1}{\beta -3}$.
An equation with roots $a$ and $b$ is $(x-a)(x-b)=0$, or $x^2-(a+b)+ab=0$.
Now
$$
a+b=\frac{2 (\alpha \beta - \alpha - \beta - 3)}{\alpha \beta - 3 \alpha - 3 \beta + 9},
\quad
ab = \frac{\alpha \beta + \alpha + \beta + 1}{\alpha \beta - 3 \alpha - 3 \beta + 9}
$$
Finally,
$$
\alpha + \beta = -\frac{1}{3},
\quad
\alpha \beta = \frac{5}{3}
$$
</s>