6. Let $f(x)$ be a quadratic function, $g(x)=2^{x} f(x), g(x+1)-g(x)=2^{x+1} \cdot x^{2}$, then $f(x)=$
6. $2 x^{2}-8 x+12$ Hint: Use the method of undetermined coefficients, we can assume $f(x)=a x^{2}+b x+c$, substitute into the conditions and proceed.