We obtain new results about the number of trinomials
${{t}^{n}}\,+\,at\,+\,b$
with integer coefficients in a box
$(a,\,b)\,\in \,[C,\,C\,+\,A]\,\times \,[D,\,D\,+\,B]$
that are irreducible modulo a prime
$p$
. As a by-product we show that for any
$p$
there are irreducible polynomials of height at most
${{p}^{1/2+o(1)}}$
, improving on the previous estimate of
${{p}^{2/3+o(1)}}$
obtained by the author in 1989.