For positive integers $n$ and $x$, let $f_n(x)$ be a polynomial in $x$ of degree $n-1$, such that $f_n(x)$ is monotonically increasing for increasing $x \ge 1$.
Now assume that there exist positive integers $k$ and $d$ such that \begin{align} f_n(16k+2) = (4k+1)d \qquad\text{and}\qquad f_n(16k+3) = (8k+1)d. \end{align}
With no other information, can anything be determined about $f_n(x)$ or $k$ or $d$?
The only interesting case is $n=3$. $n=1$ is vacuous, since there are no monotonic increasing polynomials of degree zero. For $n=2$ we can solve everything, yielding $$ f_2(x) = 4kd\,x +d(1-4k-64k^2) $$
For $n=3$, let $f_3(x) = a_2x^2+a_1s+a_0$. Then subtracting the two given equations we obtain $$ a_1 = 4kd - (32k+5)a_2 $$ which then lets us obtain $$ f_3(x) = a_2 x^2 + [4kd-(32k+5)a_2]x + [(256k^2+80k+6)a_2-d(64k^2+4k-1)] $$ So there is for a given $k$ and $d$ a 1-parameter family of polynomials that work, but we mjust start with a large enough leading coefficient, since the derivative, at $x=1$, needs to be positive: $$ a_2 > \frac{4d}{32k+7} $$
For $n>3$ the two equations seriously under-determine the $m$ coefficients, and the problem becomes uninteresting again.