Partial Fractions Decomposition I am failing to understand partial fraction decomposition in cases like the following:
Provide the partial fraction decomposition of the following:
$$\frac{x+2}{(x-4)^3(x^2 + 4x + 16)}$$
I see this and I think of 
$$\frac{A}{x-4} + \frac{Bx+C}{(x-4)^2} + \frac{Dx^2 + Ex + F}{(x-4)^3} + \frac{Gx+H}{x^2 + 4x + 16}$$
But I am told that the correct answer is
$$\frac{A}{x-4} + \frac{B}{(x-4)^2} + \frac{C}{(x-4)^3} + \frac{Dx+E}{x^2 + 4x + 16}$$
What exactly is the numerator of each fraction based on?

You have the correct understanding that the numerator is one degree lower than the denominator. For example:
$$\frac{A}{x+1}$$
$$\frac{Ax+B}{x^2+3x+2}$$
$$\frac{Ax^2+Bx+C}{x^3+2x^2+5x+1}$$
However, repeated terms are an exception. The numerator is one degree less than the repeated polynomial in the denominator. You have as many fractions as the power on the repeated term. With each fraction's denominator with a different power such that one fraction's denominator is raised to the first power, and so on raising each denominator to a higher power, until the power on the original denominator is reached. Let's look at an example:
$$\frac{x}{(x+1)^2}=\frac{A}{x+1}+\frac{B}{(x+1)^2}$$
$$\frac{x}{(x+1)^3}=\frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{C}{(x+1)^3}$$
$$\frac{x}{(x^2+x+1)^2}=\frac{Ax+B}{(x^2+x+1)}+\frac{Cx+D}{(x^2+x+1)^2}$$
