$C = bc(a+d)(b-c) + ac(b+d)(c-a) + ab(c+d)(a-b)$
$C = bc(ab - ac + db - dc) + ac(bc - ba + dc - da) + ab(ca - cb + da - db)$
$C = ab^2c - abc^2 + b^2cd - bc^2d + abc^2 - a^2bc + ac^2d - a^2cd + a^2bc - ab^2c + a^2bd - ab^2d$
$C = b^2cd - bc^2d + ac^2d - a^2cd + a^2bd - ab^2d$
$C = cd(b^2 - bc + ac - a^2) + bd(a^2 - ab) - ab^2d$
$C = cd(b(b-c) - a(a-c)) + bd(a(a-b)) - ab^2d$
$C = cd(b^2 - bc + ac - a^2) + ab(a-b)d$
$C = cd(b-a)(b+a-c) + ab(a-b)d$
$C = d[c(b-a)(b+a-c) + ab(a-b)]$
$C = d(a-b)[ab - c(b+a-c)]$
$C = d(a-b)[ab - bc - ac + c^2]$
$C = d(a-b)[b(a-c) - c(a-c)]$
$C = d(a-b)(a-c)(b-c)$