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Let's move some terms around and discuss,
Since
is always positive or zero, let call that A(x) where A(x) is always positive except when x=-1 and A(-1)=0.
Let’s talk about this function for very large positive and negative x.
For even modest values of x (x<-2 and x>0), A(x) grows rapidly because of the power of 72.
For large positive x, A(x) gives a very large, positive number, (60x-1) gives a large, positive number, therefore p(x) is a very large, positive number since positive times a positive equals a positive.
For large negative x, A(x) is a very large, positive number, (60x-1) is a large negative number, therefore p(x) is a very large, negative number since positive times a negative equals a negative.
Therefore between large positive x and large negative x comes a point where p(x)=0, that is, it has a root.
Or using the vocabulary of the IVT,
Since
There exists an x in the interval from
to
where
.
