$(A+B)(A+C)(B+C) = AB + AC + BC$ is a tautology (checked with Wolfram Alpha) and not hard to see if you apply duality principle $(invert + * 0 1)$ But how to prove with simplyfication It'S not much but here's what I have so far : $A+(BC)(B+C) = A(B+C) + BC$
And here I'm stuck.
The lefthand side should give you $(A+BC)(B+C)$, not $A+(BC)(B+C)$. That in turn expands to $AB+AC+BC+BC=AB+AC+BC$ when you ‘multiply it out’, which is exactly what you want. Your $A(B+C)+BC$, though immediately obtainable from $A+(BC)(B+C)$, is actually obtainable from $(A+BC)(B+C)$, so apparently you were working with the correct parenthesization even if you didn’t write it. In any case, with $A(B+C)+BC$ you’re essentially done: it immediately expands to $AB+AC+BC$.