find all the real rational zeros

First factor this.

x⁴ + x³ - 11x² + x - 12     Rearrange.

x³ + x + x⁴ - 11x² - 12       Factor  x  out of the first two terms and split the  -11x² .

x(x² + 1) + x⁴ + x² - 12x² - 12    Factor  x² out of  x⁴ + x²     and  -12  out of  -12x² - 12 .

x(x² + 1) + x²(x² + 1) - 12(x² + 1)     Factor  (x² + 1)  out of all the terms. 

(x² + 1)(x² + x - 12)    Factor the  x² + x - 12 .

(x² + 1)(x² + 4x - 3x - 12)

(x² + 1)(x(x + 4) - 3(x + 4))

(x² + 1)(x + 4)(x - 3)

To find out the  x  values that make this equal to zero, set each factor equal to zero and solve for  x .

x² + 1  =  0           x - 3  =  0           x + 4  =  0

x²  =  -1                 x  = 3                   x  =  -4

not real

I agree with AT.....that's a very nice factoring "trick,"  hectictar.....!!!

Well thank you!  

I did have a little hint after I saw what the answer was supposed to be from WolframAlpha though!!!

LOL!!!!!!.....some people might call that "cheating"....I prefer the term, "research"