You cannot find the unique values of two variables with only one equation. You can only establish a relationship between them.
Let z=y², then 3x⁴-7xz-2z²=4830091, which can be written:
This shows the relationship of y in terms of x and a graph of the relationship resembles a hyperbola which crosses the x axis at about ±35.62. Between these x limits, the curve doesn’t exist, so there are no values of y. Otherwise, for x>35.62 or x<-35.62, there are two values of y for each single value of x. All points on the curve are solutions to the equation.