If $x + y = -3$ and $x^2 + y^2 = 12$, what is the value of $2xy$?

The key here is to recognize that you are being asked for the value of $2xy$ and also to note that we are given the value of $x^2 + y^2$

We know that $(x+y)^2 = x^2 + y^2 + 2xy$

and therefore, $x^2 + y^2 =(x+y)^2 - 2xy.$

Thus,

$x^2 + y^2 = 12$

$(x+y)^2 - 2xy = 12.$

$(-3)^2 - 2xy = 12$

$9 - 2xy = 12$

$- 2xy = 12 - 9$

$-2xy = 3$

$2xy = -3$