A rigorous proof that is not rigorous Can we prove $x^2-1=(x+1)(x-1)$ is true for all $x$? We can do it "algebraically", but we can also do it by testing:

$x=0$ true

$x=1$ true

$x=2$ true

If it's not always true, then it's polynomial with degree $2$. But deg$(P)=$deg$(Q)=n$ and $P(a_i)=Q(a_i) \forall 1\le i\le n+1$, then $P(n)=Q(n)$ and the equation is always true.

Are there examples/explanations about "testing" if an equation is true?
You could try Mathematical Induction:

First you verify x=1 is true
Then assume that x=n is true
then prove that x=n+1 is true 
    by algebraic manipulation of x=n for your equation
    to get something equivalent to x=n+1
You could try Mathematical Induction:

First you verify x=1 is true
Then assume that x=n is true
then prove that x=n+1 is true 
    by algebraic manipulation of x=n for your equation
    to get something equivalent to x=n+1