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When faced with a cubic function, such as the one given by the expression \(h(x)=-x^3-x^2+9x+9\), it's crucial to recognize the distinctive features that categorize this type of polynomial. A cubic function is defined by having the highest exponent of 3, which results in its standard form being \(ax^3+bx^2+cx+d\), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a \eq 0\).

The most characteristic feature of cubic functions is their shape: they have a curve known as a 'sigmoid' — that is, they can resemble an elongated 'S' when sketched on a graph. Plus, they can have one, two, or three real roots (also known as zeros of the function), and these correspond to the points where the graph of the cubic function crosses the x-axis.

In the case of the equation above, the leading coefficient is negative (\(-1\)), which means the 'S' shape will be upside down, starting from the upper right and ending in the lower left of a graph. Understanding these key characteristics of cubic functions is the first step towards being able to find the zeros and sketch the graph.

Graph sketching is a valuable skill for visualizing the behavior of functions, especially for cubic functions like \(h(x)\). Even without exact calculations, certain properties can be deduced. To sketch a graph, start by plotting the zeros—points where the function intersects the x-axis. For \(h(x)\), the zeros found are at x = 0 and x = 1. With cubic functions, we additionally consider the slopes at these intercepts.

Since the coefficient of the \(x^3\) term is negative, we know the graph starts in the positive quadrant and ends in the negative quadrant. Our function doesn't cross the x-axis at the complex zeros (\(x = i\) and \(x = -i\)), but these do indicate there's a change in slope—a sort of 'twist' in the graph. By plotting turning points and inflection points, if they can be calculated easily, the graph's shape becomes clearer. The final sketch should reflect the general ‘S’ shape with consideration for the effects of any modifications caused by other terms in the function.

Factoring polynomials is a critical process when solving for zeros in functions such as the cubic function in our example. By breaking down the complex expression into simpler factors, we can more easily identify potential zeros. The goal is to find an expression that, when multiplied by itself, equals the original polynomial.

In the provided step-by-step solution, the polynomial was factored into \( -x(x^2+1)-9(x-1) = 0\). This breakdown is paramount as it reveals where the function will intersect the x-axis (at x = 0 and x = 1 for real zeros). Remember that not all cubic polynomials can be factored easily, and at times you may need to resort to other methods like synthetic division or the use of the cubic formula.

However, if factoring is possible, it reveals much about the function's structure. Any remaining factors that cannot be set to zero, as the complex ones in our function (\(x = i\) and \(x = -i\)), will simply not correspond to any real x-intercepts but still influence the graph's curvature.