How to solve this two equations simultaneously: x^2+5y^2-10y=76 and y^2=x+10 and getting four coordinates.

At the point of contact (wher ethey are simultaneous) when x and y have the same value , so hence put one into the other

so put y^2=x+10 in to x^2+5y^2-10y=76

so x=y^2-10

(y^2-10)^2 +5y^2-10y=76

y^4-20y^2+100+5y^2-10y-76=0

y^4 -15y^2-10y+24=0

let y=+1
1-15-10+24=0
so one point is y=1, x=-9...point A(-9,1)

divide (y-1) into

y^4 -15y^2-10y+24=0
leaves you
y^3+y^2-14y-24
y= -2
(-8)+4-14(-2) -24= -8+4+28-24=0
so point B y=-2, then x= y^2-10= 4-10= - 6
B(-6, -2)

dividing y+2 into y^3+y^2-14y-24
leaves
y^2-y-12=0
(y-4)(y+3)=0

at point C
y=4, x=y^2-10= 16-10= 6
(6,4) point C

y=-3, x=Y^2-10= 9-10 = -1

Point D (-1, -3)

So, the intersectiosn are
Point A (-9, 1)
Point B (-6, -2)
Point C (6, 4)
Point D (-1, -3)