Q:

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)-$3,5,7,14$.

A:

The roots have sum $-\frac{b}{a}$ and product $\frac{c}{a}$, and these sum to $\frac{c-b}{a}=\frac{b}{a}-1$. Comparing this to the sum of the roots, $(\alpha +2)(\beta +2)=3$. As the roots are integers, $\alpha, \,\beta $ are $-1,\,1$ or $-5,\,-3$ in some order. These both give a suitable quadratic, with the desired quantity either $-1$ or $7$.

