To solve these expressions, we need to perform polynomial division. Let's go through each part step-by-step:

**a)** \[(8x^6 - 4x^5 + 12x^4 - 20x^3): 4x^3\]
Divide each term in the numerator by \(4x^3\):
\[
\frac{8x^6}{4x^3} - \frac{4x^5}{4x^3} + \frac{12x^4}{4x^3} - \frac{20x^3}{4x^3} = 2x^3 - x^2 + 3x - 5
\]

**b)** \[(x^2 + 3x): x\]
Divide each term in the numerator by \(x\):
\[
\frac{x^2}{x} + \frac{3x}{x} = x + 3
\]

**c)** \[(2x^3 - x^2)\]
This is just the given expression as it is, making a note that no division is indicated here.

**d)** \[(5x^5 - x^3 - \frac{1}{2}x^2): x^2\]
Divide each term in the numerator by \(x^2\):
\[
\frac{5x^5}{x^2} - \frac{x^3}{x^2} - \frac{1}{2} \frac{x^2}{x^2} = 5x^3 - x - \frac{1}{2}
\]

So, the final results are:
a) \(2x^3 - x^2 + 3x - 5\)

b) \(x + 3\)

c) \(2x^3 - x^2\) (unchanged)

d) \(5x^3 - x - \frac{1}{2}\)