We obtain new results about the number of trinomials
with integer coefficients in a box
$(a,\,b)\,\in \,[C,\,C\,+\,A]\,\times \,[D,\,D\,+\,B]$
that are irreducible modulo a prime
. As a by-product we show that for any
there are irreducible polynomials of height at most
, improving on the previous estimate of
obtained by the author in 1989.