Having trouble finding roots/factorizing a cubic equation

Question

I've been trying to find a method that can work for any cubic equation, but I can't seem to find one. Right now, I'm trying to find the roots/ factorize the following equation: $x^3-5x^2+3x+9 = 0$. I've tried the method to see what's common in both, but does not simplify correctly. $x^2(x-5)+3(x+3)=0$

The answers are 1 and 3 with a multiplicity of 2, though I don't know how it was attained. How can I factor that equation?

Answer 1

I made a flow chart for factoring cubics for you.

Answer 2

you must searching for divisors of $9$ and we find $-1,3$ thus we have $$(x-3)^2(x+1)$$