It's not hard to intuit why this formula works, it's basically a combinatorial proof. For every summand, we select either x or the negated root from every fact

It's not hard to intuit why this formula works, it's basically a combinatorial proof. For every summand, we select either x or the negated root from every factor. This explains the alternating signs as well as the fact that adding up the exponent of x and the number of factors that go into the coefficient always results in the number of roots. Then, of course, we still need to group summands with the same exponent.