1. The problem is to identify all real roots of a function and fill in the sign table accordingly. 2. The roots given are $-5$, $0$, $2$, and $4$. These are the points where the function equals zero. 3. To fill the sign table, we analyze the sign of the function in each interval determined by these roots: $(-\infty, -5)$, $(-5, 0)$, $(0, 2)$, $(2, 4)$, and $(4, \infty)$. 4. At each root, the function changes sign if the root is of odd multiplicity; if even multiplicity, the sign does not change. 5. Since the problem does not specify multiplicities, we assume simple roots and alternate signs starting from the leftmost interval. 6. Therefore, the sign table is: - Interval $(-\infty, -5)$: positive (+) - At $x = -5$: zero (0) - Interval $(-5, 0)$: negative (-) - At $x = 0$: zero (0) - Interval $(0, 2)$: positive (+) - At $x = 2$: zero (0) - Interval $(2, 4)$: negative (-) - At $x = 4$: zero (0) - Interval $(4, \infty)$: positive (+) This completes the sign table with the correct signs and roots. Final answer: The real roots are $-5$, $0$, $2$, and $4$ with alternating signs in the intervals between them.