# How can we simplify $(a\pm b)(c\pm d)$?

Is there any way to simplify the expression:

$$(a\pm b)(c\pm d)\quad ?$$ At first, I tried just distributing the values, and through FOIL I achieved:

$$ac \pm ad \pm bc \pm bd$$

This works when the signs multiply to positive, but if one of the signs is negative and the other positive, it gives the wrong lower bound.

The problem is that the plus-or-minus bd "depends" on the signs of b and d.

Is there a concise and succinct way to write this expression?

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The sequence of symbols $$(a\pm b)(c\pm d)$$ is an expression, not an equation (there's no equals sign). – Zev Chonoles Feb 3 '13 at 3:08

Using two $\pm$ signs in an expression is inherently ambiguous, because at least in some contexts and according to some conventions, they are assumed to be linked, i.e. either both plus or both minus.
If you're really set on writing everything in one equation, I'd recommend $$(a+(-1)^rb)(c+(-1)^sd)=ac+(-1)^rbc+(-1)^sad+(-1)^{r+s}bd$$ and then say whatever it is you intend about $r$ and $s$.