How do I multiply [ p-1/2][p+1/2]?

FOIL method. First, outer, inner, then last terms.

You'll get p^2 +1/2p - 1/2p - 1/4 = p^2 - 1/4

[p - 1/2][p + 1/2]
= p*p - 1/2*p + p*1/2 - 1/2*1/2
= p^2 + p/2 + p/2 - 1/4
= p^2 - 1/4

foil
first
outside
inside
last

for example
(a+b)(a+b)
aa+ab+ab+bb
aa+2ab+bb

p² + (1/2)p - (1/2)p - 1/4
p² - 1/4

foil, (first-outisde-inside-last)