How would you calculate a trinomial that cannot be factored?

What do you mean to calculate?

If it can't be factored, it's simplified.

x² + x + 1

It's an expression that can't be simplified further.  There's nothing to calculate.  What are you wanting to do with it?

comment : in theory every polynomial can be factored....find the roots , form the factors ( x - root)...multiply them together , multiply this answer by the ' leading coefficient '....x² + x + 1 = ( x + 1/2)² + 3/4 = ( x + 1/2 + i√(3/4) ) ( x + 1/2 - i√(3/4))