The number of irrational roots of $(x^2 -3x +1)(x^2 +3x +2)(x^2 -9x + 20) = -30$ is___? 

The number of irrational roots of $(x^2 -3x +1)(x^2 +3x +2)(x^2 -9x + 20) = -30$ is___?



My Approach : 
Multiplying the above equation and then apply Descartis rule is very lengthy method. Also I tried it by taking factors of $30$ and then solving the equation but that is also going very lengthy . 
Please suggest some good method of doing this.

I'm not sure if this is a good method for your question, but we have
$$(x^2-3x+1)(x+1)(x+2)(x-4)(x-5)=-30$$
rearranging
$$(x^2-3x+1)(x+1)(x-4)(x+2)(x-5)=-30$$
$$(t+1)(t-4)(t-10)=-30$$
where $t=x^2-3x$.
Now, one sees that $t=5$ works. So, 
$$(t+1)(t-4)(t-10)+30$$
is divisible by $t-5$ to have
$$(t+1)(t-4)(t-10)+30=(t-5)(t^2-8t-14)$$
So,
$$x^2-3x=t=5,4\pm\sqrt{30}$$
