Use the intermediate value theorem to show that the polynomial function has a real zero between the numbers given.
\[
x^{4}-4 x^{3}-20 x^{2}+32 x+59 ;-2 \text { and }-1
\]

Step 1 ：Define the polynomial function \(f(x) = x^{4} - 4x^{3} - 20x^{2} + 32x + 59\).

Step 2 ：Calculate the function values at -2 and -1, denoted as \(f(-2)\) and \(f(-1)\) respectively.

Step 3 ：\(f(-2) = -37\) and \(f(-1) = 12\).

Step 4 ：Since \(f(-2)\) and \(f(-1)\) have different signs, it means that the function must cross the x-axis (i.e., have a zero) somewhere between -2 and -1.

Step 5 ：By the Intermediate Value Theorem, the polynomial function \(f(x) = x^{4} - 4x^{3} - 20x^{2} + 32x + 59\) has a real zero between -2 and -1.

Step 6 ：\(\boxed{\text{Final Answer: The polynomial function } x^{4} - 4x^{3} - 20x^{2} + 32x + 59 \text{ has a real zero between -2 and -1.}}\)