Since the square of any number is always positive, $b^2 > 0$
Since $b^2c < 0$, or a positive number * c = negative number, that means that c must be a negative number.
$abc > 0$, so abc is a positive number.
abc = ab * (negative number) = positive number
That means that ab must be a negative number, since the product of two negative numbers is positive.
Therefore, ab < 0, so Quantity B is greater.

To demonstrate, say that $a = 1$, $b = -1$, and $c = -2$.
$b^2c = (-1)^2(-2) = -2 < 0$
$abc = (1)(-1)(-2) = 2 > 0$
$ab = (1)(-1) = -1 < 0$

OR $a = -1$, $b = 1$, and $c = -2$.
$b^2c = (1)^2(-2) = -2 < 0$
$abc = (-1)(1)(-2) = 2 > 0$
$ab = (-1)(1) = -1 < 0$