Which of the binomials below is a factor of this trinomial ? x^2 + x + 12

That trinomial can't be factored. You would have to complete the square to factor it and even then your answer would be an imaginary number. 

x² + x + 12 = 0
x² + x = - 12
x² + x + 1/4 = - 12 + 1/4
x² + x + 1/4 = - 11 3/4
x² + x + 1/4 = - 47/4

(x + 1/2)² = - 47/4 
x + 1/2 = +/- √(- 47/4) 
x + 1/2 = +/- 

x = 1/2 +/- 

If the constant was a negative, it would be factor-able (is that a word?). 

Multiplication property of zero