Let a ∈ R and let α, β be the roots of the equation x ^ 2 +...

Since, $\alpha, \beta$ be the roots of the equation ${x}^{2} + 60\dfrac{1}{4}x + a = 0$, therefore

$\alpha + \beta = - 60\dfrac{1}{4}$

$\alpha\beta = a$

Given:

${\alpha}^{4} + {\beta}^{4} = - 30$

$⇒ {\left({\alpha}^{2} + {\beta}^{2}\right)}^{2} - 2{\alpha}^{2}{\beta}^{2} = - 30$

$⇒ {\left\{{\left(\alpha + \beta\right)}^{2} - 2\alpha\beta\right\}}^{2} - 2{a}^{2} = - 30$

$⇒ {\left\{60\dfrac{1}{2} - 2a\right\}}^{2} - 2{a}^{2} = - 30$

$⇒ 60 + 4{a}^{2} - 4a \times 60\dfrac{1}{2} - 2{a}^{2} = - 30$

$⇒ 2{a}^{2} - 4 \cdot 60\dfrac{1}{2}a + 90 = 0$

$⇒ {a}^{2} - 2 \cdot 60\dfrac{1}{2}a + 45 = 0$

Product of roots$= 45$