Problem

(a) Use the binomial expansion to expand

$$(4-5x)^{-\frac{1}{2}}\qquad |x|<\frac{4}{5}$$

in ascending powers of $x$, up to and including the term in $x^2$ giving each coefficient as a fully simplified fraction.

$$f(x)=\frac{2+kx}{\sqrt{4-5x}}\qquad \text{where }k\text{ is a constant and }|x|<\frac{4}{5}$$

Given that the series expansion of $f(x)$, in ascending powers of $x$, is

$$1+\frac{3}{10}x+mx^2+\cdots\qquad \text{where }m\text{ is a constant}$$

(b) find the value of $k$,

(c) find the value of $m$.