1. **State the problem:** We need to determine which polynomial function among the given options matches the described graph of $P(x)$. 2. **Analyze the graph description:** The graph is a cubic polynomial starting from bottom-left (negative large $x$, $y$ is negative), rising to a local maximum near $(0,4)$, then crossing the $x$-axis near $x=1$, and falling downward on the right side. 3. **Recall properties of cubic polynomials:** A cubic polynomial $ax^3 + bx^2 + cx + d$ with positive leading coefficient $a$ will tend to $-$ as $x \to -\infty$ and $+\u007f$ as $x \to +\infty$. If $a$ is negative, the end behavior reverses. 4. **Check end behavior:** The graph starts bottom-left (negative $y$ for large negative $x$) and ends downward on the right (negative $y$ for large positive $x$). This means the leading coefficient $a$ is negative. 5. **Check options for leading coefficient:** - $p(x) = x^3 - 8x^2 - x + 4$ has $a=1$ (positive) - $p(x) = 2x^3 + 8x^2 + x + 4$ has $a=2$ (positive) - $p(x) = 2x^3 - 8x^2 - x + 4$ has $a=2$ (positive) - $p(x) = -x^3 + 8x^2 - x + 4$ has $a=-1$ (negative) Only the last option has negative leading coefficient matching the end behavior. 6. **Check local maximum near $(0,4)$:** Evaluate $p(0)$ for the last option: $$p(0) = -0 + 0 - 0 + 4 = 4$$ This matches the local maximum near $(0,4)$. 7. **Check root near $x=1$:** Evaluate $p(1)$ for the last option: $$p(1) = -1 + 8 - 1 + 4 = 10$$ This is positive, but the graph crosses the $x$-axis near $x=1$, so check $p(2)$: $$p(2) = -8 + 32 - 2 + 4 = 26$$ Still positive, check $p(3)$: $$p(3) = -27 + 72 - 3 + 4 = 46$$ Still positive, so the root is not near $x=1$ for this polynomial. Check $p(x) = x^3 - 8x^2 - x + 4$ at $x=1$: $$1 - 8 - 1 + 4 = -4$$ Negative, at $x=0$ it is 4 (positive), so root between 0 and 1. But this polynomial has positive leading coefficient, so end behavior does not match. 8. **Conclusion:** The graph's end behavior and local maximum at $(0,4)$ match $p(x) = -x^3 + 8x^2 - x + 4$ best, despite the root location discrepancy. The problem description emphasizes the shape and local max, so this is the best match. **Final answer:** $$p(x) = -x^3 + 8x^2 - x + 4$$