Best way to solve inequality with multiple parameters

Questions
,
1

I tried to solve the inequality

b^2 c^2 - 4 c^3 - 27 d - 4 b^3 d + 18 b c d > 0

using

FindInstance[b^2 c^2 - 4 c^3 - 27 d - 4 b^3 d + 18 b c d > 0, Reals]

but I get the error

"The system contains a nonconstant expression b independent of \
variables {\!\(\*TemplateBox[{},\"Reals\"]\)}."

Any idea why, or alternative method to solve the inequality?

2

I checked the documentation page for FindInstance usage

FindInstance Docs

and tried

FindInstance[b^2 c^2-4 c^3-27 d-4 b^3 d+18 b c d>0,{b,c,d}, Reals]

which instantly returned

{{b -> -(29^(1/3)/2^(2/3)), c -> 0, d -> 1}}
3

You get a full symbolic solution with Reduce:

Reduce[b^2 c^2 - 4 c^3 - 27 d - 4 b^3 d + 18 b c d > 0,
 {c, b, d}, Reals]