When first differences contradict a regular regression regarding Investment vs Output relationship
It's hard to say. Presumably you think you have some fixed (unobservable) omitted variables, which is why you ran fixed effects (FE) / first differences. So I see one of two things going on here:


The fixed omitted variables caused significance in the first regression, which means you do not have a significant effect. Do you have fixed omitted variables? If not, go with the first regression instead.
A quadratic model fits the data much better and hence you have significance in the first model, but not the second. In that case, you probably do have a significant effect. Is that true though? Why did you choose a quadratic specification? Does theory or the scatter plot suggest that is a good idea? If not go with the second regression. 


My suggestion would be to try fixed effects/first differences  with the qudratic terms if you have good reason for both. Otherwise use the specification you think has the most merit.

Unfortunately, I am not familiar with R and cannot comment on the code.