If graph of $y = ax^2 -bx + c$ is following, then sign of $a$, $b$, $c$ are

Suppose the quadratic polynomial $p(x)=a x^2+b x+c$ has positive coefficient $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$ then what could be the possible value of $\alpha+\beta+\alpha \beta$ if $0 \leq \alpha+\beta+\alpha \beta \leq 8$

Let $S=\left\{ x : x \in R \text { and }(\sqrt{3}+\sqrt{2})^{ x ^2-4}+(\sqrt{3}-\sqrt{2})^{ x ^2-4}=10\right\} \text {. }$ Then $n ( S )$ is equal to

lf $2 + 3i$ is one of the roots of the equation $2x^3 -9x^2 + kx- 13 = 0,$ $k \in R,$ then the real root of this equation

The number of real solutions of the equation $e ^{4 x }+4 e ^{3 x }-58 e ^{2 x }+4 e ^{ x }+1=0$ is..........