F(x)= 9x^3 + 10x^2 - 17x - 2 one zero given is -2

Given one zero of the polynomial function, find the other zeros.

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We begin with the goal of the question. It gives a 'zero', by which I assume it means the x-intercept where y = 0. We seek to find the other x-intercepts (points where the graph crosses the x-axis and thus points whose y coordinate is zero).
Since this is a third degree polynomial, it will have three roots, through which we find values of x (although it's possible values repeat and there can only be one point with the same coordinates, so it can have three or fewer x-intercepts).
In general, in these nice cases, we find the intercepts using the knowledge that, at that point, f(x) will be equal t zero. We also know that anything multiplied by zero will be zero, so we attempt to factor the polynomial into its roots. For example, g(x) = x^2 -1 factors into g(x) =(x-1)(x+1). To make g(x) = 0, either the first or second part must equal zero, so x-1=0 and x+1=0, and thus x =1 and x = -1 are the x-coordinates where g(x) will be equal to zero. We use the same principle for third degree polynomials.
We have the first coordinate, x = -2, so we work backwards to find that x+2=0. This means that (x+2) is one factor of 9x^3 + 10x^2 - 17x - 2. Here comes the tricky part: we have to factor it out. This is done by polynomial long division.
Remember how regular long division works? Set up the thing to be divided (9x^3 + 10x^2 - 17x - 2) to the left and box/block off the thing to be divided by (x+2) to the right. Make sure the polynomials are in proper order, with left-to-right descending exponents of x. Now look at the leftmost (largest exponent of x) term to divide (in this case 9x^3) and the leftmost term of the thing to be divided by (here, x) and divide (9x^3 / x = 9x^2). Put this as the first term in your answer. Now take this term (9x^2) and multiply the thing you're dividing by by it ( 9x^2 * (x+2) = 9x^3 + 18x^2). Write this under the polynomial you're dividing, aligning each term below the term with same x-exponent. Now subtract and you have a new polynomial, but since you've canceled the terms with the largest x-exponent, your new polynomial will be of a lower degree. :D