For an idea to why just think about what happens when x gets very large or very small. When either happens the leading term dominates the expression. For example for a cubic polynomial things grow cubically much faster than quadratically so similarly because things grow faster for higher degree terms when approaching higher values (infinity but Id rather not mention infinity) the leading term dominates the expression. So analyzing -4x^5 - 3x^3 +4x -20 is really like analyzing -4x^5 and when large enough the 4 doesn't make much of a "difference" either so its like analyzing (-1)x^5. This is why you just look at the leading coefficient to see if it is positive or negative as the actual value of the constant doesn't really change how infinity large something is. Then once you know the associated polynomial to analyze that is simple you just look at its end behavior instead of your original polynomial, i.e. -x^5 is positive towards negative infinity and negative at positive infinity (odd function).
