Answer: \(\frac1{256}-\frac{x^2}{24}+\frac{7x^4}{36}\); coefficient \(\frac{25}{576}\).

(a) In the expansion of

\(\left(\frac12-\frac23x^2\right)^8,\)

the first three terms come from \(r=0,1,2\).

For \(r=0\),

\(\left(\frac12\right)^8=\frac1{256}.\)

For \(r=1\),

For \(r=2\),

So the first three terms are

\(\frac1{256}-\frac{x^2}{24}+\frac{7x^4}{36}.\)

(b) Also,

\(\left(2x+\frac1x\right)^2=4x^2+4+\frac1{x^2}.\)

The coefficient of \(x^2\) comes from three products:

Therefore the coefficient of \(x^2\) is

\(\frac4{256}-\frac4{24}+\frac7{36}.\)

Using denominator \(576\),