Fully Simplifying a Boolean Expression with DeMorgan's Theorem
\$\overline{(\bar{A}B+\bar{A}C})+(BC + \bar{B}\bar{C})\Leftrightarrow \$ 

You made a mistake with the first \$\bar{B}\$ and the operator should also be inverted:

\$\\(\overline{\bar{A}B}\cdot\overline{\bar{A}C})+(BC + \bar{B}\bar{C}) \Leftrightarrow\$ 

Now you can apply De Morgan's law again

\$\\((\bar{\bar{A}}+\bar{B})\cdot(\bar{\bar{A}}+\bar{C}))+(BC + \bar{B}\bar{C}) \Leftrightarrow\$ 

\$\\((A+\bar{B})\cdot(A+\bar{C}))+(BC + \bar{B}\bar{C}) \Leftrightarrow\$ 

\$\\(AA+\bar{B}A+A\bar{C}+\bar{B}\bar{C})+(BC + \bar{B}\bar{C}) \Leftrightarrow\$ 

\$\\(A+\bar{B}\bar{C})+(BC + \bar{B}\bar{C}) \Leftrightarrow\$ 

\$\\A+\bar{B}\bar{C}+BC + \bar{B}\bar{C} \$ 

To not finish your homework (probably) , you can continue from here.