If f ( x ) = g ( x ) h ( x )does the...

Let $F:x\mapsto ax+b$b be the linear approximation of $f$ around $0$, and $G:x\mapsto cx+d$ be the one of $g$. Then we have $FG:x\mapsto (ax+b)(cx+d)=ac{x}^2+(bc+ad)x+bd$ which is clearly not linear.
But actually the linear approximation of $fg$ is $x\mapsto (bc+ad)x+bd$ since you just need to get rid of the $x^2$ term.
For quadratic approximations, it's the same : you can multiply the two approximations, but then you need to get rid of all the terms with $x^k$ where $k>4$.
except for some particular cases, it is false.