12.5.1 Olympic algebra

Problem: Find the integer part of the largest value of the parameter \( a \), for which the equation \[ 3 \log _{5} \frac{x^{2}+a x+4}{x}+5\left(x^{2}+(a-1) a+4\right)=0 \] has positive roots and all these roots are integers. If the desired value doesn't exist, write the word "no" as an answer.