\(y[x] = \(x\^p\) \((a\  + \ b\ x\  + c\ x\^2)\); 
yp = D[y[x], x]; 

    ypp = D[y[x], {x, 2}]; 
Series[E\^\(2\ x\), {x, 0, 8}]\)

Expand[y[x] (ypp^2)]