Question

How do i tell if this trinomial 8x^2 - 14x-3 is prime or not?

Answer 1

Is it prime over the integers, the rationals or the real numbers?

Answer 2

your equation
\[
8 x^2-14 x-3=0
\]
Has two real roost but they are not rationals. Here they are
\[
\left\{\left\{x= \frac{1}{8}
\left(7-\sqrt{73}\right)\right\},\left\{x=
\frac{1}{8}
\left(7+\sqrt{73}\right)\right\}\right\}
\]
So your polynomial is prime over the rationals, but not prime over the reals.