Ok this is a great problem to work with. Let me show you two types of examples here:
Example 1: x2+3x+2
Step 1: Factor this problem out! --------> x2+3x+2 = (x+1)(x+2)
Step 2: Now you know from multiplication rule that if one of the sums within the parenthesis equals 0, then the whole product of the two parenthesis would be 0. So you equal each parenthsis to 0.
(x+1) = 0 -------> x=-1
(x+2) = 0 -------> x=-2
Notice that when x = -1 the expression
(x+1)(x+2)= (-1+1)(-1+2) = (0)(1)=0
that when x = -2 the expression
(x+1)(x+2)= (-2+1)(-2+2) = (-1)(0)=0
Tadaa! So when a problem asks to get zeros of a polynomial you are just finding the x values where the polynomial expression would equal 0. The fastest way to do it is through factoring.
Example 2: x2+2x+3
Sometimes you cannot factor. So you should use the quadratic formula for this. You can use the quadratic formula for example 1 but it's just faster to factor.