To simplify the expression \((x^2 - x + 20)^2 + (x - 5)^2(x^2 - x + 20)^2 + (x - 5)^2\), we can first notice some common factors.

Let's denote \( A = (x^2 - x + 20)^2 \) and \( B = (x - 5)^2 \). Thus, the expression can be rewritten as:

\[
A + B A + B
\]

Factoring out \( A \) from the first two terms, we get:

\[
A(1 + B) + B
\]

Now, substituting back for \( A \) and \( B \):

\[
(x^2 - x + 20)^2(1 + (x - 5)^2) + (x - 5)^2
\]

Next, let's simplify \( 1 + (x - 5)^2 \):

\[
1 + (x - 5)^2 = 1 + (x^2 - 10x + 25) = x^2 - 10x + 26
\]

So we can rewrite the expression as:

\[
(x^2 - x + 20)^2(x^2 - 10x + 26) + (x - 5)^2
\]

At this point, we can leave the expression in this form, or further expand \( (x^2 - x + 20)^2 \) and then multiply it out with \( (x^2 - 10x + 26) \), but that results in extensive calculations.

Overall, we have simplified the expression to:

\[
(x^2 - x + 20)^2(x^2 - 10x + 26) + (x - 5)^2
\]

This is a suitable stopping point. To summarize:

\[
\boxed{(x^2 - x + 20)^2(x^2 - 10x + 26) + (x - 5)^2}
\]