Smooth values of polynomials

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\epsilon > 0$,thereareinfinitelymany$n\in$\mathbb{N}$$ for which the largest prime factor of $f(n)$ is no larger than $n^{ϵ}$.