Parabola directrix formula

Question

Question 1: I don't understand why (p+x)^2 = (MP)^2 Does it include positive and negative values?

Question 2: Why is x + p = 0

Any help is much appreciated. Thank you.

Answer 1

It's clear from the picture that when the focus is $p$ units to the right of the origin and the directrix $p$ units to the left that the parabola lives in the right half plane, so $x \ge 0$. Then $p+x$ is the horizontal distance from the directrix to $M$.

The answer to your second question is that $x = -p$ is the equation of the vertical line through $(-p, 0)$. Since the line is vertical there's no $y$ in that equation.