Find the points of intersection between the line y=x+7 and the curve x^2+2x+5.Also it asks to sketch the area between the graph

Points of intersection:

y = x+7
y = x^2 + 2x +5

Equate both equations

x+7 = x^2 + 2x +5

x^2 + x -2 = 0

Through factorisation: (x-1)(x+2) = 0   Hence, x = 1,-2.

Let's substitute our values of x into one of our above equations.

y=x+7

If x=1, then y=6,  if x=-2, then y=5

Therefore, there are 2 points of intersection, (1,6) and (-2,5).

Graph plot: http://www.flickr.com/photos/82005243@N00/195261288/

I haven't shaded their area however. I'm sure you can do that.