How can you say it is true or false? Can we compare an equation like I did. Given that $p(x)$ has real roots. If it true, Would it also be true when roots are imaginary?
Is it true. Please help
It's not quite right. Saying $p(x)$ has two roots $\alpha$ and $\beta$ does not mean that $p(x)=(x-\alpha)(x-\beta)$. You are forgetting that any constant times $(x-\alpha)(x-\beta)$ will still have roots $\alpha$ and $\beta$. So, what you can conclude is that
$$
p(x)=c(x-\alpha)(x-\beta),
$$
where $c$ is a constant. From the expansion, it follows, in fact, that $c=a$.
To prove your statement wrong, it suffices to show a single counter-example.

$$p(x):=2x^2-6x+4$$ and $$q(x):=(x-1)(x-2)$$ both have the roots $1$ and $2$, yet $a\ne1$.