a. $(x-5y)(5y+x)$
$= (x-5y)(x+5y)$
$= x^2-(5y)^2$
$= x^2-25y^2$
b. $(m+n)^2+(m-n)^2$
$= m^2+2mn+n^2+m^2-2mn+n^2$
$= 2m^2+2n^2$
c. $2(x-y)(x+y)+(x+y)^2+(y-x)^2$
$= (x+y)^2+2(x-y)(x+y)+(x-y)^2$
$= (x+y+x-y)^2$
$= 4x^2$
d. $2x(x+3)-2(x-1)(x+1)$
$= 2x^2+6x-2(x^2-1)$
$= 2x^2+6x-2x^2+2$
$= 6x+2$
$= 2(3x+1)$