Condition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$ Find the set of values of $a$ if $$(x^2+x)^2+a(x^2+x)+4=0$$ has
$(i)$ All four real and distinct roots
$(ii)$ Four roots in which only two roots are real and distinct.
$(iii)$ All four imaginary roots
$(iv)$ Four real roots in which only two are equal.
Now if I set $x^2+x=t$ then even if $t^2+at+4=0$ has real roots in is not necessary that $(x^2+x)^2+a(x^2+x)+4=0$ will have real roots too.  So how to derive the condition on a?
Could someone give me some direction?

Alternatively, graphing it:
Let $a=y$, then $(x^2+x)^2+y(x^2+x)+4=0$
$\displaystyle y=\frac{1}{4}-\left( x+\frac{1}{2} \right)^{2}+
\frac{4}{\frac{1}{4}-\left( x+\frac{1}{2} \right)^{2}}$

