Given that (x+y+z)(xy+xz+yz)=25 and that x^2(y+z)+y^2(x+z)+z^2(x+y)=7 for real numbers x, y, and z, what is the value of xyz?

(x+y+z)(xy+xz+yz)=25,  x^2(y+z)+y^2(x+z)+z^2(x+y)=7 

Expand the first equation 

yx^2 + zx^2 + xyz  + xy^2 + xyz + zy^2  + xyz + xz^2 + yz^2   =  25  simplify

yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2 + 3xyz = 7     (1) 

Expand the second equation

yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2  = 7      (2)

Substitute  (2) into (1)

7 + 3xyz  =   25       subtract 7 from 25

3xyz   = 18            divide both sides by 3

xyz  = 6