Actually, this question should contain "...solution between x=2.2 and 2.3".
Substitute x=2.2 in x^3+8x-30=-1.752
Now substitute x=2.3: x^3+8x-30=0.567.
The sign of the first result is negative so x=2.2 in x^3+8x gives an answer <30 and the sign of the second result is positive so x=2.3 gives the answer >30. That proves that x must be in between 2.2 and 2.3. (The answer is approximately 2.2761.)