Relationship between roots 
Suppose we have the equation $x^2 + 2x + 1, x^2 + 3x + 1, \ldots, x^2 + (2 + k) x + 1, k \in \mathbb{N}$, consider all the roots less than $-1$. (see Appendix)
    Is it possible to find the length of any one of the neighboring segments, knowing one of them? For example, if I know the length of a red segment, can I only use the knowledge of the equation and the length of the segment to find the blue segment?


Define
$$f_k(x):=x^2+(2+k)x+1$$
for $x<-1$.
When $f_k(x)=0$,
$$
x=\frac{-(2+k)\pm\sqrt{(2+k)^2-4}}{2}
$$
Since $k+2>2$, $$(2+k)^2-4>0$$ so we have two distinct roots. We can show that the case where $x<-1$ is when
$$x=-\frac{(2+k)+\sqrt{(2+k)^2-4}}{2}$$
Define $$r_k:=-\frac{(2+k)+\sqrt{(2+k)^2-4}}{2}$$
Define $$l_k:=r_{k}-r_{k+1}$$
as the lengths between the adjacent roots of $y=f_k$ and $y=f_{k+1}$.
We have
\begin{align}
l_k
&=\frac{(3+k)+\sqrt{(3+k)^2-4}}{2}-\frac{(2+k)+\sqrt{(2+k)^2-4}}{2}\\
&=\frac12\left(1+\sqrt{(3+k)^2-4}-\sqrt{(2+k)^2-4}\right)
\end{align}
which you can then use to calculate back your unknown $k$ from your known $l_k$ to then calculate $l_{k+1}$.
