a quadratic equation for two unknown number find values of $p$ such that the equation $4x^2 + 3px - 2p = 0$ has? below are a few choices of the value p:
a) 2 real roots
b) 1 real roots
c) no roots or complex roots

so far i did for a) 2 real roots
$$(3p)^2 - 4(4)(-2p) > 0\\
9p^2 +32p > 0\\
9\left( p^2+\frac{32p}{9} \right) > 0\\
p^2+\frac{32p}{9} > 0\\
p^2 +\frac{32p}{9} + \left(\frac{32}{18}\right)^2 > 0\\
\left(p + \frac{32}{18} \right) ^2 > 0 \\
p + \frac{32}{18} > 0\\
p > -\frac{32}{18} $$

$D=(3p)^2+4\cdot4\cdot2p=9p^2+32p=p(9p+32)$
if $D>0$ then there is 2(different) real roots
$D=9p^2+32p=p(9p+32)>0$
$p\in(-\infty;-32/9)\cup(0;+\infty)$
if $D=0$ then there is 1(or 2 equal) real root
$p=0$ and $p=-32/9$
if $D<0$ then there is no roots or 2 complex roots
$p\in(-32/9;0)$
