How could I know that 4x^4 + 3^2 - 3 has two roots?
You don't. It's not true. Or are you asking about real roots?
Is that really the problem? so so that there are four roots, the four fourth roots of -3/2, none of which are real.
If the equation is , we can let so we are dealing with the quadratic equation . By the quadratic formula,
which has the two roots y= 6/8= 3/4 and y= -8/8= -1.
Now, If , then , two real roots. If , then , two imaginary roots.
The crucial point was that, because this had only even factors, we could reduce to a second order equation. Positive roots of that equation give real roots of the original equation.