Step 1: Identify the function and the table structure. The function is $Y = 4 + 3x - x^2$. The table requires calculating values for $4$, $3x$, $-x^2$, and $Y$ for $x$ values from $-1$ to $5$. Step 2: Complete the calculation for $x=0$. The previous response was cut off while calculating for $x=0$. For $x = 0$: $$Y = 4 + 3(0) - (0)^2$$ $$Y = 4 + 0 - 0$$ $$Y = 4$$ Step 3: Calculate the values for $x=1, 2, 3, 4, 5$. For $x = 1$: $$Y = 4 + 3(1) - (1)^2$$ $$Y = 4 + 3 - 1$$ $$Y = 6$$ For $x = 2$: $$Y = 4 + 3(2) - (2)^2$$ $$Y = 4 + 6 - 4$$ $$Y = 6$$ For $x = 3$: $$Y = 4 + 3(3) - (3)^2$$ $$Y = 4 + 9 - 9$$ $$Y = 4$$ For $x = 4$: $$Y = 4 + 3(4) - (4)^2$$ $$Y = 4 + 12 - 16$$ $$Y = 0$$ For $x = 5$: $$Y = 4 + 3(5) - (5)^2$$ $$Y = 4 + 15 - 25$$ $$Y = -6$$ Step 4: Complete the table. The table is completed by calculating $4$, $3x$, $-x^2$, and $Y = 4 + 3x - x^2$ for each $x$ value. | $x$ | $4$ | $3x$ | $-x^2$ | $Y = 4 + 3x - x^2$ | |-----|-----|------|--------|----------------------| | $-1$ | $4$ | $-3$ | $-1$ | $0$ | | $0$ | $4$ | $0$ | $0$ | $4$ | | $1$ | $4$ | $3$ | $-1$ | $6$ | | $2$ | $4$ | $6$ | $-4$ | $6$ | | $3$ | $4$ | $9$ | $-9$ | $4$ | | $4$ | $4$ | $12$ | $-16$ | $0$ | | $5$ | $4$ | $15$ | $-25$ | $-6$ | The completed table is: $$\boxed{ \begin{array}{|c|c|c|c|c|} \hline x & 4 & 3x & -x^2 & Y = 4 + 3x - x^2 \\ \hline -1 & 4 & -3 & -1 & 0 \\ \hline 0 & 4 & 0 & 0 & 4 \\ \hline 1 & 4 & 3 & -1 & 6 \\ \hline 2 & 4 & 6 & -4 & 6 \\ \hline 3 & 4 & 9 & -9 & 4 \\ \hline 4 & 4 & 12 & -16 & 0 \\ \hline 5 & 4 & 15 & -25 & -6 \\ \hline \end{array} }$$