$x + y = 7 $and $x^3 - y^3 = 37$, find $xy$ 
$x+y=7$
  $x^3-y^3 = 37$
Find $xy$

I have tackled this one but I am stuck, when I get to this point:
$$(x-y)((x+2)^2 -xy) = 37$$
I would be glad if you could give me any suggestions.

Since $(x^3-y^3)^2$ is symmetric in $x$ and $y$, you can express it as a polynomial in terms of $x+y$ and $xy$ :
$(x^3-y^3)^2 = x^6 - 2x^3y^3 + y^6 = (x+y)^6 - (6x^5y+15x^4y^2+22x^3y^3+15x^2y^4+6xy^5) \\
= (x+y)^6 - 6xy(x+y)^4 + (9x^4y^2+14x^3y^3+9x^2y^4) \\
= (x+y)^6 - 6xy(x+y)^4 + 9(xy)^2(x+y)^2 - 4(xy)^3$
Plugging the values $x+y = 7$ and $x^3-y^3 = 37$ gives you the equation
$4(xy)^3 - 441(xy)^2 +14406(xy)-116280 = 0$
The only rational root of this cubic polynomial is $12$, so this factors as 
$(xy-12)(4(xy)^2-393(xy)+9690) = 0$
And now you can check that the second factor doesn't have real roots.
Finally, getting a value for $xy$ (here, $12$) determines the values of $x$ and $y$ completely, because
$x = (x+y)/2 + (x^3-y^3)/(2((x+y)^2-xy))= 7/2 + 37/(2.(7^2-12)) = 4$ and
then $y = (x+y)-x = 7-4 = 3$
