De Morgan minimization
De Morgan
$$y=\overline{a+\overline{b(\overline{c+d})}}+\bar{b}$$
$$y=\bar{a}(b(\overline{c+d}))+\bar{b}$$
You are correct up to this point.  Based on the comments, you are correct for the rest, but you fooled up the mathjax syntax as you deMorganed.  Corrected we have:
$$y=\bar{a}b\bar c \bar d+\bar{b}$$

You are also headed in the correct direction:
$$X+\bar{X} Y = X + Y$$  

Which is the redundancy law.  The inclusion of \$X\$ means the \$\bar X\$ in \$\bar X Y\$ is redundant.