17. Let $f(x)$ be a quadratic function, and $f(0)=12$
$$
\begin{array}{c}
g(x)=2' f(x) . \\
g(x+1)-g(x) \geqslant 2^{x+1} x^{2} .
\end{array}
$$

Find the expressions for $f(x)$ and $g(x)$.
17. Since $f(x)$ is a quadratic function, it can be set as
$$
f(x)=a x^{2}+b x+12(a \neq 0) .
$$

Substituting (1) into (2) gives

that is,
$$
\begin{array}{c}
2^{n+1} f(x+1)-2^{x} f(x) \geqslant 2^{x+1} x^{2}, \\
2 f(x+1)-f(x) \geqslant 2 x^{2} .
\end{array}
$$

Substituting (3) into (4), we get
$$
a x^{2}+(4 a+b) x+(2 a+2 b+12) \geqslant 2 x^{2} .
$$

(1) $a=2, b=-8$.
Thus, $f(x)=2 x^{2}-8 x+12$,
$$
g(x)=\left(2 x^{2}-8 x+12\right) 2^{x} .
$$

Verification shows that this is the solution.
Or (2) when $a \neq 2$, $\left\{\begin{array}{l}a>2 \\ \Delta=(4 a+b)^{2}-4(a-2)(2 a+2 b+12) \leqslant 0 .\end{array}\right.$
That is,
$$
\begin{array}{c}
8(a+1)^{2}+(b-16)^{2} \leqslant 168(a>2) . \\
2<a \leqslant \sqrt{21-\frac{1}{8}(b-16)^{2}}-1 .
\end{array}
$$

In summary, $f(x)=2 x^{2}-8 x+12 \cdot g(x)=\left(2 x^{2}-8 x+12\right) 2^{x}$ or $f(x)=a x^{2}+b x+12, g(x)=$ $\left(a x^{2}+b x+12\right) 2^{x}$, where $a$ and $b$ satisfy $2<a \leqslant \sqrt{21-\frac{1}{8}(b-16)^{2}}-1$.

Explanation: If the structure of the function can be predicted, then the function can be expressed using undetermined coefficients, reducing the problem to determining the coefficients. This is usually done by solving a system of equations to find the specific values of the undetermined coefficients.