multiplicity just tells you how many times a root is "repeated".

for instance, if we had a quadratic of the form , the only root would be , since this makes the quadratic zero. but, a quadratic must have two roots. so there should be one more, but low and behold! the square tells us the root is repeated a second time. so is a root with multiplicity 2

note that

what x's make this zero?

note that

what x's make this zero?

h(x) is already in a nice factored form, what x's make it zero? the powers tell you the multiplicity

for p(x), look at the graph. wherever the graph touches the x-axis, that is a root. if it passes through, the multiplicity is 1, if there is a hump and the graph touches the x-axis without cutting it, the multiplicity is 2.