Identify the conic section? Is my answer correct?

NOTE that your earlier hyperbola had a minus sign between the squared terms

and

The ellipse as a positive sign.   If the numbers on  the botton were the same then it would be a circle. 

A circle is a special case of an elipse because it has a double focii instead of 2 separate focii. 

Effectively a cirlce has one centre and other ellipses have 2 centres.

HERE IS THE GRAPH :)

https://www.desmos.com/calculator/yd3iwle8y9

$$\\4x^2 + 7y^2 + 32x - 56y + 148 = 0 \\\\ 4x^2+ 32x + 7y^2 - 56y =-148 \\\\ 4(x^2+8x)+7(y^2-8y)=-148\\\\ 4(x^2+8x+16)+7(y^2-8y+16)=-148+4*16+7*16\\\\ 4(x+4)^2+7(y-4)^2=28\\\\ \frac{4(x+4)^2}{28}+\frac{7(y-4)^2}{28}=1\\\\ \frac{(x+4)^2}{7}+\frac{(y-4)^2}{4}=1\\\\ \frac{(x+4)^2}{(\sqrt7)^2}+\frac{(y-4)^2}{2^2}=1\\\\$$

This is an ellipse centre (-4,+4)

The ends of the major axis are   $$(-4-\sqrt7,4) and (-4+\sqrt7,4)$$

Then ends of the minor axis are  (-4,4-2) (-4,4+2)  that is    $$(-4,2) and (-4,6)$$

references

http://www.mathsisfun.com/geometry/ellipse.html

http://en.wikipedia.org/wiki/Ellipse

Oh and Welcome to the forum Tyler :)