Answer:

[tex] \boxed{\bf \: \bigg( \dfrac{ \alpha }{ \beta } + 2\bigg) \bigg(\dfrac{ \beta }{ \alpha } + 2 \bigg) = \dfrac{q + 2 { p}^{2} }{q}} \\ [/tex]

Step-by-step explanation:

Given that,

[tex]\sf \: \alpha \: and \: \beta \: are \: zeroes \: of \: polynomial \: {x}^{2} + px + q \\ [/tex]

We know,

[tex]\boxed{{\sf Sum\ of\ the\ zeroes=\dfrac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}} \\ [/tex]

[tex]\sf \:  \implies \: \alpha + \beta = - \dfrac{p}{1} = - p \\ [/tex]

Also,

[tex]\boxed{{\sf Product\ of\ the\ zeroes=\dfrac{Constant}{coefficient\ of\ x^{2}}}} \\ \\ [/tex]

[tex]\sf \:  \implies \: \alpha \beta = \dfrac{q}{1} = q\\ [/tex]

Now, Consider

[tex]\sf \: \bigg( \dfrac{ \alpha }{ \beta } + 2\bigg) \bigg(\dfrac{ \beta }{ \alpha } + 2 \bigg) \\ [/tex]

[tex]\sf \: = \: \bigg( \dfrac{ \alpha + 2 \beta }{ \beta }\bigg) \bigg(\dfrac{ \beta + 2 \alpha }{ \alpha } \bigg) \\ [/tex]

[tex]\sf \: = \: \dfrac{ \alpha \beta + {2 \alpha }^{2} + {2 \beta }^{2} + 4 \alpha \beta }{ \alpha \beta } \\ [/tex]

[tex]\sf \: = \: \dfrac{ \alpha \beta + 2( { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta ) }{ \alpha \beta } \\ [/tex]

[tex]\sf \: = \: \dfrac{ \alpha \beta + 2{ (\alpha + \beta )}^{2}}{ \alpha \beta } \\ [/tex]

[tex]\sf \: = \:\dfrac{q + 2 {( - p)}^{2} }{q} \\ [/tex]

[tex]\sf \: = \:\dfrac{q + 2 { p}^{2} }{q} \\ [/tex]

Hence,

[tex]\implies\sf \: \boxed{\bf \: \bigg( \dfrac{ \alpha }{ \beta } + 2\bigg) \bigg(\dfrac{ \beta }{ \alpha } + 2 \bigg) = \dfrac{q + 2 { p}^{2} }{q}} \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information:

[tex]\bf \:If\:\alpha, \beta \: are \: zeroes \: of \: {ax}^{2} + bx + c, \: then \\ [/tex]

[tex]\sf \: \alpha + \beta = - \dfrac{b}{a} \\ [/tex]

[tex]\sf \: \alpha \beta = \dfrac{c}{a} \\ [/tex]

[tex]\bf \:If\: \alpha, \beta, \gamma \: are \: zeroes \: of \: {px}^{3} + {qx}^{2} + rx + s, \: then\\ [/tex]

[tex]\sf \: \alpha + \beta + \gamma = - \dfrac{q}{p} \\ [/tex]

[tex]\sf \: \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{r}{p}\\ [/tex]

[tex]\sf \: \alpha \beta \gamma = - \dfrac{s}{p} \\ [/tex]