(A ∪ B) ∩ (B ∩ C')

Substitute the sets for the letters:

({2, 4, 6} ∪ {1, 2, 5, 8}) ∩ ({1, 2, 5, 8} ∩ {1, 3, 7}')

As in algebra, we work inside parentheses first:

First we find this complement set {1, 3, 7}' whose elements are
all the elements of U = {1, 2, 3, 4, 5, 6, 7, 8} which are NOT
elements of {1, 3, 7}.  That's {2, 4, 5, 6, 8}. So
replace {1, 3, 7}' by {2, 4, 5, 6, 8}, and now we have:

({2, 4, 6} ∪ {1, 2, 5, 8}) ∩ ({1, 2, 5, 8} ∩ {2, 4, 5, 6, 8})

Next we find this union {2, 4, 6} ∪ {1, 2, 5, 8} by taking all
the elements that are elements of either {2, 4, 6} or {1, 2, 5, 8},
or both.  That's the set {1, 2, 4, 5, 6, 8}.  So we replace 
{2, 4, 6} ∪ {1, 2, 5, 8} by {1, 2, 4, 5, 6, 8} and now we have: 

{1, 2, 4, 5, 6, 8} ∩ ({1, 2, 5, 8} ∩ {2, 4, 5, 6, 8})

Next we find the intersection {1, 2, 5, 8} ∩ {2, 4, 5, 6, 8}
by taking only the elements which are contained in both
{1, 2, 5, 8} and {2, 4, 5, 6, 8}.  That's the set {2, 5, 8}
So we replace {1, 2, 5, 8} ∩ {2, 4, 5, 6, 8} by {2, 5, 8}

{1, 2, 4, 5, 6, 8} ∩ {2, 5, 8}

One more step. We find the intersection 
{1, 2, 4, 5, 6, 8} ∩ {2, 5, 8} by taking only the elements 
which are contained in both {1, 2, 4, 5, 6, 8} and {2, 5, 8}
That's the set {2, 5, 8}. So that's the final answer:

{2, 5, 8}

Edwin