$\displaystyle \alpha$ and $\displaystyle \beta$ are zeroes of a quadratic polynomial $\displaystyle x^{2}-a x-b$. Obtain a quadratic polynomial whose zeroes are $\displaystyle 3 \alpha+1$ and $\displaystyle 3 \beta+1$.

\[\alpha+\beta=a, \alpha \beta=-b \] Sum of zeroes of required polynomial \[=(3 \alpha+1)+(3 \beta+1) \] \[\begin{aligned} & =3(\alpha+\beta)+2 \\ & =3 a+2 \\ & \text { Product of zeroes of required polynomial } \\ & =(3 \alpha+1)(3 \beta+1) \\ & =9 \alpha \beta+3(\alpha+\beta)+1 \\ & =-9 b+3 a+1 \\ & \therefore \text { The required polynomial is } x^{2}-(3 a+2) x+(3 a-9 b+1) \end{aligned} \]