In my textbook, I was given the problem: 

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> Finds the roots of the equation $3x^4 + 4x^3 + x - 1 = 0$  
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>  a. three positive real numbers and one negative real number  
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>  b. three negative real numbers and one positive real number  
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>  c. one negative real number and three imaginary numbers  
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>  d. one positive real number, one negative real number, and two imaginary numbers  
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>  e. two positive real numbers, one negative real numbers, and one imaginary number
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By looking at the graph of the function, you see that it passes through the x-axis on both the positive and negative sides, so you know that it must have a positive and negative real root. Given the choices here, you can then deduce that the answer (which is correct) is 

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> d. one positive real number, one negative real number, and two imaginary numbers  
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simply because if imaginary roots must come in conjugate pairs if there are no imaginary coefficients in the simplified form of the equation. However, the textbook's reasoning confuses me:

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> Observe that the graph crosses the x-axis twice - once at a positive x value and once at a negative one. Since the function is a degree 4 polynomial, there are 4 roots, so the other two must be complex conjugates. 
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The explanation makes sense for the most part, but I am confused as to why one would be able to discredit the function from having 2 pairs of double real roots. This is not one of the options to this question, so this isn't the case here, but I'm curious if there is a way to tell if something has double real roots or not (in a timed environment, such as a test, because I am already aware of one of the long ways to do this: factoring and using synthetic division to find all the roots). 

I tried this out with the equation $(x-2)^4$. I was thinking that if you wanted to test if 2 was a root, you would plug it in $(2-2)^4$ and see the result is 0, so it is a factor (according to the factor theorem). I attempted to continue this logic and figured that in $(x-2)^4$, 2 is a factor 4 times, so I attempted to plug in $((2^4)-2)^4$. However, off the bat I could see this idea wouldn't work. I am wondering if there is a quick way to tell if something has double, quadruple, etc. real roots or if something has complex conjugate pairs of imaginary roots.