To find the zeros and intercepts of the quadratic function $f(x) = 3x^2 + 10x - 8$:

Finding the zeros: Set the function equal to zero, $0 = 3x^2 + 10x - 8$. Factor the trinomial to obtain $(x + 4)(3x - 2) = 0$. Apply the zero product property by setting each factor to zero ($x + 4 = 0$ or $3x - 2 = 0$), which yields the solutions $x = -4$ and $x = \frac{2}{3}$.

Finding the x-intercepts: An x-intercept occurs when the function's value is zero ($y = 0$). Since $f(-4) = 0$ and $f(\frac{2}{3}) = 0$, the x-intercepts of the graph are the points $(-4, 0)$ and $(\frac{2}{3}, 0)$.

Finding the y-intercepts: A y-intercept occurs when $x = 0$. Substitute $0$ for $x$ to find $f(0) = 3(0)^2 + 10(0) - 8$, which simplifies to $-8$. Therefore, the y-intercept is the point $(0, -8)$.