The problem you provided asks to find the units digit of $p(2^{2021})$, given the following polynomial relation:

$2p(x) + 2x^2 - p(x^2 - 2x + 4) + 8x + 5 = 0$

Let's solve this step by step.

Step 1: Simplify the given expression.

We are provided the equation:

$2p(x) + 2x^2 - p(x^2 - 2x + 4) + 8x + 5 = 0$

Rearranging terms:

$2p(x) - p(x^2 - 2x + 4) = -2x^2 - 8x - 5$

This gives us a functional relationship between $p(x)$ and $p(x^2 - 2x + 4)$.

Step 2: Analyze specific values.

We are tasked with finding the units digit of $p(2^{2021})$, but to proceed, we need to either determine the exact form of the polynomial $p(x)$ or find patterns in its behavior based on simpler inputs like $x = 0, 1, 2$, etc.

I will now perform detailed calculations to extract further information about the polynomial form or its behavior mod 10, as the problem likely relates to finding patterns in the units digits.

Let me proceed with that.It seems there was an issue with defining the function properly. I'll redefine the function and retry simplifying the equation. Let's first focus on extracting the form of the polynomial $p(x)$.

I'll make sure to handle the setup correctly this time.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to attempt the problem step by step manually, or would you prefer trying again at a later time?

Here are five follow-up questions you might consider:

1. What happens if you plug in specific values for $x$ to simplify the equation?
2. How would you identify patterns in polynomial functions when specific terms are given?
3. Can we deduce the behavior of $p(x)$ modulo 10 for specific values of $x$?
4. What role does modular arithmetic play in simplifying large exponentiations like $2^{2021}$?
5. How could functional equations help solve unknown polynomials?

Tip: Solving polynomial equations often requires exploring patterns for small values of the variables involved.