Computing the value of $(x+y)^4$ if $x^4+y^4=5$ and $x^2+xy+y^2=10$ 
Let $x^4+y^4=5$ and $x^2+xy+y^2=10.$ Find $(x+y)^4.$

First, I tried expanding $(x+y)^4$ using the binomial theorem to get $5+4x^3y+6x^2y^2+4xy^3,$ so simplifying I got $5+4xy(x^2+y^2)+6(xy)^2.$ Then I rearranged the given equation to get $x^2+y^2=10-xy,$ so the expansion becomes $5+4xy(10-xy)+6(xy)^2.$ I further simplified to get $2x^2y^2+40xy+5,$ but I'm not sure how to continue off here. Factoring this expression doesn't seem to help. May I have some help? Thanks in advance.

We have $95 = (x^2+xy+y^2)^2 - (x^4+y^4) = 2x^3 y + 2xy^3 + 3x^2 y^2,$ which you may recognize. Any further and I'd just give away the entire solution.
