The given expression is P = |a-b| + 3(a-c)^2 + 5(c-b)^4 + 4.

To simplify this expression, let's break it down step by step:

Step 1: |a-b|
The absolute value of the difference between a and b can be positive or negative depending on their values. To simplify this, we can consider two cases:

Case 1: a ≥ b
In this case, |a-b| = a - b.

Case 2: a < b
In this case, |a-b| = -(a - b) = b - a.

Therefore, we can simplify |a-b| as max(a-b, b-a).

Step 2: 3(a-c)^2
This term is already simplified and cannot be further simplified.

Step 3: 5(c-b)^4
This term is also already simplified and cannot be further simplified.

Now, let's put all the simplified terms together:

P = max(a-b, b-a) + 3(a-c)^2 + 5(c-b)^4 + 4

Note: The simplified expression may vary depending on the values of a, b, and c.