Properties from Exponential or Trigonometric form Q8

Given that \(z=r\left( \cos \theta +\mathrm{i}\sin \theta \right)\), solve for \(z\) in the form \(x+y\mathrm{i}\) if \(4+\mathrm{i}\left( 4z+1 \right)=2r{{e}^{\mathrm{i}\left( \pi +\theta \right)}}\).