Find the equation of a quadratic with vertex at (−4,−5), passing through the point (−3,−7).?

Starting from the general vertex form:

y = a(x - h)² + k

Where the vertex is (h, k).  This is (-4, -5) in your case so we can substitute these values into h and k:

y = a(x + 4)² - 5

We are then given another point (-3, -7).  We can use these values for x and y:

-7 = a(-3 + 4)² - 5

We now have one unknown left that we can solve for:

-7 = a(1)² - 5

-7 = a(1) - 5

-7 = a - 5

-2 = a

Now your equation is:

y = a(x + 4)² - 5

y = -2(x + 4)² - 5

If you want this in polynomial form, expand the right side:

y = -2(x² + 8x + 16) - 5

y = -2x² - 16x - 32 - 5

y = -2x² - 16x - 37

The equation of a quadratic with vertex at (−4,−5), 

passing through the point (−3,−7)

y = -2 x^2 - 16 x - 37