If a is an integer and $a\in (-5,30]$ then the probability that the graph of the function $y=x^2+2(a+4)x-5a+64$  is strictly above the x-axis is

$\begin{array}{rcl}\mathrm{a}& \in & \left[-5, 30\right]\\ \mathrm{n}\left(\mathrm{S}\right)& =& 35\end{array}$

E= given graph of function

$\mathrm{y}={\mathrm{x}}^2+2(\mathrm{a}+4) \mathrm{x}-5\mathrm{a}+64$ is Strictly above x-axis

$\Rightarrow$ 'Q' Eq has no real roots

$$\begin{gathered} \Rightarrow ∆<0 \Rightarrow 4{(\mathrm{a}+4)}^2 -4\left(1\right) (-5\mathrm{a}+64)<0 \\ \Rightarrow {\mathrm{a}}^2+16+8\mathrm{a}+5\mathrm{a}-64<0 \\ \Rightarrow (\mathrm{a}+16) (\mathrm{a}-3)<0 \\ \Rightarrow -16<\mathrm{a}<3 \\ \mathrm{a}=\left\{-4, -3, -2, -1, 0, 1, 2\right\} \\ \mathrm{n}\left(\mathrm{E}\right)=7 \Rightarrow \mathrm{P}\left(\mathrm{E}\right)=\frac{7}{35}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right. \end{gathered}$$