1. **Problem:** Determine if the function $f(x) = x^4 - 5x^3 + x^2 + 12x - 5$ has a zero between 0 and 1. 2. **Method:** Use the Intermediate Value Theorem which states that if a continuous function changes sign over an interval, it must have a zero in that interval. 3. **Evaluate at $x=0$:** $$f(0) = 0^4 - 5\cdot0^3 + 0^2 + 12\cdot0 - 5 = -5$$ 4. **Evaluate at $x=1$:** $$f(1) = 1 - 5 + 1 + 12 - 5 = (1 - 5) + 1 + 12 - 5 = -4 + 1 + 12 - 5 = 4$$ 5. Since $f(0) = -5$ (negative) and $f(1) = 4$ (positive), the function changes sign between 0 and 1. 6. **Conclusion:** By the Intermediate Value Theorem, there is at least one zero of $f(x)$ in the interval $(0,1)$. **Final answer:** Yes, $f(x)$ has a zero between 0 and 1.