Find the area of the region included between the circle: x2 + y2 + 2 x + 10y + 27 = 0 and the parabola y2 + 10y – x + 23 = 0

Check and make sure there is not a typo in the circle equation. Maybe that should be -27?
As is, it has a negative radius which is infeasible.

Assuming there was a typo with the circle equation, I am going to go with

x^{2}+y^{2}+2x+10y-27=0

Completing the square, we get:

(x+1)^{2}+(y+5)^{2}=53

The circle has center at (-1,-5) and has radius \sqrt{53}\approx 7.28


Solving this for y, we get:

y=-\sqrt{-x^{2}-2x+52}-5 and

y=\sqrt{-x^{2}-2x+52}-5

The parabola can be solved for y and we get:

y=-\sqrt{x+2}-5

\sqrt{x+2}-5

Now, use this info, along with some integration, to find the area.

Please use ^ to designate powers. x2 looks like x times 2.