$4\left(x^{2} − 6x + 9\right) − \left(y^{2} + 10y + 25\right) = 5 + 36 − 25$

$4\left(x − 3\right)^{2} − \left(y + 5\right)^{2} = 16$

$\frac{(x-3)^2}{4}-\frac{(y+5)^2}{16}=1$

This is a hyperbola.

See part (a).

$\left(x + 5\right)^{2} + \left(y − 2\right)^{2} = 16$
This is a circle.

See part (a).

$\frac{(x+3)^2}{9}+\frac{(y-2)^2}{4}=1$

This is an ellipse.

Set the equation equal to y and complete the square for x.

$y = 3\left(x − 2\right)^{2} + 5$
This is a parabola.