<s>[INST] Quadratic polynomial with integer roots. The quadratic polynomial $ax^2+bx+c$ has positive coefficients $a,b,c$ in A.P. in the given order. If it has integer roots $\alpha,\beta,$ find $\alpha+\beta+\alpha \beta.$
I tried with Vieta's theorem and putting $b=\frac{a+c}{2}$ to get $\alpha+\beta+\alpha \beta=\frac{b}{a}-1=\frac{c-a}{2a}$ but couldn't arrive at a solution.
P.S. The question had the following options given of which one and only one is the correct answer (if they are of any help)-$3,5,7,14$.
 [/INST] Unpacking J.G.'s answer for the masses:
$ax^2+bx+c$
$$\text{if} \quad (a,b,c) \quad \text{are in "AP", then} \quad  $$
$$\begin{align}\begin{cases}
a&=a \\
b&=a+d \\
c&=a+2d 
\end{cases}
\ \ &\underbrace{\implies}_{\text{b is the average of a and c}} \ \ 
\left[\frac{a+c}{2} =\frac{2a+2d}{2}=a+d=b\right]
\\
&\qquad \quad  \implies \qquad \quad \ \left[c=2b-a\right]\end{align}$$
Vieta:  if $\alpha$ and $\beta$ are the roots $\implies \frac{-b}{a}=\alpha + \beta$ and $\frac{c}{a}=\frac{2b-a}{a}=\alpha \beta $
$$\begin{align}
\implies \alpha+\beta+\alpha \beta&=\frac{c-b}{a} \\
&=\frac{b-a}{a} \\
&=\frac{b}{a}-1 \\
&=-(\alpha+\beta)-1 \\
\alpha \beta +2(\alpha+\beta)&=-1 \\
\alpha \beta +2(\alpha +\beta)+4&=3 \\
(\alpha+2)(\beta+2)&=3=d_1 \cdot d_2 \\
\text{such that}\quad (d_1,d_2) &\in \{(1,3),(3,1),(-1,-3),(-3,-1) \\
&\begin{cases}
\alpha=d_1-2 \\
\beta=d_2-2
\end{cases} \\
\text{so} \quad (\alpha,\beta)&\in\{(-1,1),(1,-1),(-3,-5),(-5,-3) \\
\implies \alpha + \beta + \alpha \beta&=-1 \quad \lor \quad 7
\end{align}$$ 
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