The above figure represents the motion of a particle at P, where the position vector are represented by
$\vec{r_{p}}$ - position vector of the point P with respect to the inertial frame.
$\vec{r_{o'}}$ - position vector of the intermediate reference frame wrt to the inertial.
$\vec{r_{p/o'}}$ - position vector of point P wrt to the intermediate reference frame.
$\omega = \omega_m $ - angular velocity about the intermediate reference frame
$\alpha$ - the angular acceleration about the intermediate reference frame
then the velocity and acceleration can be derived as follows.
$\vec{r_p} = \vec{r_{o'}} + \vec{r_{p/o'}}$
The general material derivative for fixed reference frame is give as shown below.
$\dfrac{D ()}{Dt} = \dfrac{d ()_{o'}}{d t} + \omega_m \times ()_{p/o'} + \dfrac{d ()_{p/o'}}{d t} $
So, the velocity of point P with respect to the inertial reference frame will be
$ \vec{v_{p}} = \vec{v_{o'}} + \omega_m \times \vec{r_{p/o'}} + \vec{v_{p/o'}} $
and so, the acceleration is obtained by taking the first derivative,
$ \vec{a_p} = \dfrac{D{\vec{v_{p}}}}{Dt} $
$ \qquad = \dfrac{D \vec{v_{o'}}}{Dt} + \dfrac{D (\omega \times \vec{r_{p/o'}})}{Dt} + \dfrac{D \vec{v_{p/o'}}}{Dt} $
$\qquad = \vec{a_{o'}} + \omega \times \vec{v_{o'}} + \dfrac{D(v_{o'})}{Dt}_{| Pure~translation} +\dfrac{D \omega}{Dt} \times \vec{r_{p/o'}} + \omega \times \dfrac{D \vec{r_{p/o'}}}{Dt} + \dfrac{D\vec{v_{p/o'}}}{Dt} $
$ \qquad = \vec{a_{o'}} + \omega \times \vec{v_{o'}} + \dfrac{D(v_{o'})}{Dt}_{| Pure~translation} + \vec{\alpha} \times \vec{r_{p/o'}} + (\omega \times \omega) \times \vec{r_{p/o'}} + \dfrac{D \vec{\omega}}{Dt}_{|pure~translation} \times \vec{r_{p/o'}} + \vec{\omega} \times \vec{v_{p/o'}} +\omega \times (\vec{\omega} \times \vec{r_{p/o'}} ) + \omega\times \dfrac{D(\vec{r_{p/o'}})}{Dt}_{|pure~ translation} + \dfrac{D\vec{v_{{p/o'}_{pure~translation}}}}{Dt} $
what is wrong in the above derivation for the correct answer should be
$ \vec{a_p} = \vec{a_{o'}} + \vec{\alpha} \times \vec{r_{p/o'}} + 2 \omega \times \vec{v_{p/o'}} + \omega \times (\omega \times \vec{r_{p/o'}}) + \dfrac{D \vec{v_{p/o'}}}{Dt} $