(a) 求

按 $x$ 的升幂展开时的前四项，并将每一项写成最简形式。

(b) 求该展开式成立时 $x$ 的取值范围。

先把原式改写成 $2(1+u)^{-2}$，其中 $u=-5x/2$。先简化二项式系数，再代入 $u$，避免把多个未简化乘积放在同一行。

Let

With $u=-5x/2$, this becomes

For exponent $-2$, the coefficients are

Hence

With $u=-5x/2$,

Multiplying each term by $2$, the first four terms are

也可以直接对 $(2-5x)^{-2}$ 展开，不必先提出常数因子。分别求四项后再乘 $8$；得到的展开与解法一相同。

Expand $8(2-5x)^{-2}$ directly.

For exponent $-2$, the binomial coefficients are

and

Use these with successive powers of $2$ and $-5x$.

The constant and linear terms are

The quadratic and cubic terms are

Multiplying each term by $8$, the required expansion begins

负整数指数的二项式展开要求变量部分的绝对值小于 $1$。这里变量部分是 $-5x/2$，解对应的绝对值不等式即可。

The expansion is valid when

Hence

Therefore the required range is