1. The problem involves analyzing the polynomial function $$f(x) = (x - 3)(x + \sqrt{5})(x - \sqrt{5})$$ and understanding its properties. 2. First, expand the function by multiplying the factors step-by-step. 3. Note that $$(x + \sqrt{5})(x - \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5$$ by the difference of squares formula. 4. Substitute back to get $$f(x) = (x - 3)(x^2 - 5)$$. 5. Now expand this product: $$f(x) = x(x^2 - 5) - 3(x^2 - 5) = x^3 - 5x - 3x^2 + 15$$. 6. Rearrange terms in descending powers of $x$: $$f(x) = x^3 - 3x^2 - 5x + 15$$. 7. This is the fully expanded polynomial function. 8. To find the x-intercepts, set $$f(x) = 0$$: $$(x - 3)(x + \sqrt{5})(x - \sqrt{5}) = 0$$. 9. The solutions are $$x = 3, x = -\sqrt{5}, x = \sqrt{5}$$. 10. These are the points where the graph crosses the x-axis. 11. The function is a cubic polynomial with leading coefficient 1, so it behaves like $$x^3$$ for large $$|x|$$. 12. The graph will cross the x-axis at the three roots found. Final answer: $$f(x) = x^3 - 3x^2 - 5x + 15$$ with x-intercepts at $$x = 3, x = -\sqrt{5}, x = \sqrt{5}$$.