If f(x)=(-x3-x+5)/(2x3+3x2-7) the domain has only one constraint which is when the denominator=0. Otherwise the domain is from minus infinity to plus infinity.
2x3+3x2-7=0 has only one real zero (the other two are complex).
When x=1, 2x3+3x2-7=-2, and when x=2, 2x3+3x2-7=16+12-7=21.
This means that there is a zero between x=1 and x=2. This zero will be irrational (so check the question to make sure that the given expression is correct).
Various methods can be used to find the zero (it appears to be quite close to x=1).
One method is to use the Intermediate Value Theorem (IVT). Since the zero is between 1 and 2, we can then try x=1.5: 2x3+3x2-7=6.5 (still positive), so the zero is between 1 and 1.5. Next value to try is x=1.25, which gives the result 1.59 approx, still positive. Next value is x=1.125, which gives a negative result, so the zero is between x=1.125 and x=1.25, so x=1.1875, giving a positive result. And so on, till we get to x=1.149376 approx when the result is almost zero. This is the excluded value in the domain. All other values of x are valid.