1) $\cdots g(x)=2^{x+1} \cdot x^{2}, x \in \mathbf{R}$. Find the expression for $g(x)$.
Analysis Let $f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)$. Then

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$$
2^{x+1}\left(a(x+1)^{2}+b(x+1)+c\right)-2^{x}\left(a x^{2}+b x+c\right)=2^{x+1} \cdot x^{2},
$$

Therefore, for any $x \in \mathbf{R}$, we have $2 a(x+1)^{2}+2 b(x+1)+2 c-a x^{2}-b x-c=2 x^{2}$, which is
$$
a x^{2}+(4 a+b) x+2 a+c=2 x^{2} \text {, }
$$

By comparing the coefficients of each power of $x$ on both sides, we get: $a=2, b=-8, c=-4$, thus,
$$
g(x)=2^{x+1}\left(x^{2}-4 x-2\right) \text {. }
$$