Answer: \(a=2\), \(b=-3\), \(c=480\).

First expand the part \((a+bx)^5\) as far as the \(x^2\) term:

\((a+bx)^5=a^5+5a^4bx+10a^3b^2x^2+\cdots.\)

Now multiply by \(1+x\). The constant term is

\(a^5.\)

The coefficient of \(x\) is

\(a^5+5a^4b.\)

The coefficient of \(x^2\) is

\(10a^3b^2+5a^4b.\)

Comparing with

\(32-208x+cx^2,\)

gives

\(a^5=32.\)

Therefore

\(a=2.\)

Next,

\(a^5+5a^4b=-208.\)

Using \(a=2\),

\(32+80b=-208.\)

So

\(b=-3.\)

Finally,

\(c=10a^3b^2+5a^4b.\)

Substitute \(a=2\) and \(b=-3\):

\(c=10(8)(9)+5(16)(-3)=720-240=480.\)