if alpha nad beta are zeroes of x2+px+q then find the value of (alpha/beta+2)(beta/alpha+2)

Since $\alpha \mathrm{and} \beta$ are the roots of the polynomial $x^2+px+q = 0$
So sum of the roots = $\alpha +\beta = -p ...\left(\mathrm{i}\right)$
Product of the roots = $\alpha \beta = q ...\left(\mathrm{ii}\right)$
So we have;
$$\begin{gathered} \left(\frac{\alpha }{\beta }+2\right)\left(\frac{\beta }{\alpha }+2\right) \\ = \frac{\alpha }{\beta }\times \frac{\beta }{\alpha }+\frac{2\alpha }{\beta }+\frac{2\beta }{\alpha }+4 \\ = 1+\frac{2{\alpha }^2+2{\beta }^2}{\alpha \beta }+4 \\ = 1+\frac{2\left({\alpha }^2+{\beta }^2\right)}{\alpha \beta }+4 \\ = 5+\frac{2\left[{\left(\alpha +\beta \right)}^2-2\alpha \beta \right]}{\alpha \beta } \\ = 5+\frac{2\left[{\left(-p\right)}^2-2q\right]}{q} \{\mathrm{using} (\mathrm{i}) \mathrm{and} (\mathrm{ii}\left)\right\} \\ = \frac{5q+2p^2-4q}{q} \\ = \frac{q+2p^2}{q} \end{gathered}$$