Factoring x^5+x+1 over a finite field using modulo arithmetic

[amcavoy]

How can I factor $$x^5+x+1$$ using modulo? I know, for example, I could write $$(x+3)(x+5)$$ as $$x^2+x+1 mod7$$. How can I go backwards with this?

Thanks.

 

[matt grime]

You can't.