One way to prove a solution is between two numbers is to prove that one number is negative and the other is positive. This means the graph must cross 0 between the points and thus there is a solution.
So all we have to do is plug in 1 and 2 for x and prove one is negative and the other is positive.
If we plug in 1, we get -3. If we plug in 2, we get positive 10. This means there is a solution!