(a) Find the zeroes of the polynomial 4 x^2+4 x -3 and verify the relationship between zeroes and coefficients of the polynomial.

\[\begin{aligned} & \mathrm{P}(\mathrm{x})=4 x^{2}+4 x-3 \\ & \quad=(2 \mathrm{x}+3)(2 \mathrm{x}-1) \\ & \therefore \text { Zeroes of the polynomial are } \frac{-3}{2}, \frac{1}{2} \\ & \text { Sum of Zeroes }=\frac{-3}{2}+\frac{1}{2}=\frac{-3+1}{2}=-1=\frac{-4}{4}=\frac{-(\text { coefficient of } \mathrm{x})}{\left(\text { coefficient of } \mathrm{x}^{2}\right)} \\ & \text { Product of Zeroes }=\frac{-3}{2} \times \frac{1}{2}=\frac{-3}{4}=\frac{\text { constant term }}{\text { coefficient of } \mathrm{x}^{2}} \end{aligned} \]

Here \(\displaystyle \alpha+\beta=-1\) and \(\displaystyle \alpha \beta=-2\)

\[\begin{aligned} \frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^{2}+\beta^{2}}{\alpha \beta} & =\frac{(\alpha+\beta)^{2}-2 \alpha \beta}{\alpha \beta} \\ & =\frac{(-1)^{2}-2(-2)}{-2}=-\frac{5}{2} \end{aligned} \]