The given expression is:

\[ x^2 - 2xy + y^2 - z^2 + 2zt + t^2 \]

To simplify or factor this expression, we can look for patterns or common algebraic identities. Let's break it down step by step.

First, notice that the terms \( x^2 - 2xy + y^2 \) form a perfect square trinomial:

\[ x^2 - 2xy + y^2 = (x - y)^2 \]

Next, observe that the terms \( -z^2 + 2zt + t^2 \) also form a perfect square trinomial:

\[ -z^2 + 2zt + t^2 = -(z^2 - 2zt + t^2) = -(z - t)^2 \]

So, we can rewrite the original expression as:

\[ (x - y)^2 - (z - t)^2 \]

This is now in the form of a difference of squares, which can be factored using the identity \( a^2 - b^2 = (a - b)(a + b) \):

Let \( a = (x - y) \) and \( b = (z - t) \). Then,

\[ (x - y)^2 - (z - t)^2 = [(x - y) - (z - t)][(x - y) + (z - t)] \]

Simplifying the terms inside the brackets:

\[ [(x - y) - (z - t)] = (x - y - z + t) \]
\[ [(x - y) + (z - t)] = (x - y + z - t) \]

Therefore, the factored form of the given expression is:

\[ (x - y - z + t)(x - y + z - t) \]

So, the simplified and factored form of \( x^2 - 2xy + y^2 - z^2 + 2zt + t^2 \) is:

\[ (x - y - z + t)(x - y + z - t) \]