#### Answer

$b=36$ or $b=-36$

#### Work Step by Step

RECALL:
There are two forms of perfect square trinomials:
(1) $m^2+2mn+n^2$, which is the square of $(m+n)^2$; and
(2) $m^2-2mn+n^2$, which is the square of $(m-n)^2$
The given trinomial has:
$m^2 = 4z^2=(2z)^2$, which means that $m=2z$
$m^2=81=9^2$, which means that $n=9$
Thus, the given trinomial will be a perfect square if
(i) $bz=2mn=2(2z)(9) = 36z$, which means that $b=36$; and when
(ii) $bz=-2mn=-2(2z)(9) = -36z$, which means that $b=-36$
Therefore, the given polynomial is a perfect square when $b=36$ and when $b=-36$