The answer is yes. You can do it using Fast Fourier Transform(FFT) to make FFT you need $n \log^k n$ operations. You not really need to know what FFT do. You need only to know that you can evaluate polynomial at some $n$ points using one FFT and you can calculate the polynomial from its values at $n$ points in one FFT.
The following algorithm should work: Using FFT evaluate $F(aX+b)$ at n points it is the same as to calculate F(X) at $n$ points. Next again using FFT you can calculate $F(aX+b)$.
