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Answer :
We start with the synthetic division table where the left number is $5$, and the coefficients of the dividend are written as:
$$
\begin{array}{cccc}
5\,| & 2 & 10 & 1 \\
& & -10 & 0 \\
\hline
& 2 & 0 & 1 \\
\end{array}
$$
Even though the number on the left is $5$, the multiplication step uses $-5$ (since $2 \times (-5) = -10$). This indicates that the divisor is actually
$$
x - (-5) = x + 5.
$$
Now, follow these steps:
1. **Bring down the first coefficient:**
The first coefficient is $2$, so it remains unchanged.
2. **Multiply by the divisor’s associated number:**
Multiply $2$ by $-5$ to obtain $-10$.
3. **Add to the next coefficient:**
Add $-10$ to the second coefficient $10$, resulting in $0$.
4. **Multiply again by $-5$:**
Multiply the result $0$ by $-5$ to get $0$.
5. **Add to the last coefficient:**
Add $0$ to the third coefficient $1$, resulting in a remainder of $1$.
The bottom row now shows the quotient coefficients and the remainder:
- **Quotient coefficients:** $[2, 0]$, which represents the quotient polynomial $2x + 0$ (simply $2x$).
- **Remainder:** $1$.
By the polynomial division algorithm, we have:
$$
\text{Dividend} = (\text{Divisor})(\text{Quotient}) + \text{Remainder}.
$$
Substitute the known values:
$$
\text{Dividend} = (x + 5)(2x) + 1.
$$
Now, expand the product:
$$
(x + 5)(2x) = 2x^2 + 10x.
$$
Finally, adding the remainder:
$$
2x^2 + 10x + 1.
$$
Thus, the dividend polynomial is:
$$
\boxed{2x^2 + 10x + 1}.
$$
$$
\begin{array}{cccc}
5\,| & 2 & 10 & 1 \\
& & -10 & 0 \\
\hline
& 2 & 0 & 1 \\
\end{array}
$$
Even though the number on the left is $5$, the multiplication step uses $-5$ (since $2 \times (-5) = -10$). This indicates that the divisor is actually
$$
x - (-5) = x + 5.
$$
Now, follow these steps:
1. **Bring down the first coefficient:**
The first coefficient is $2$, so it remains unchanged.
2. **Multiply by the divisor’s associated number:**
Multiply $2$ by $-5$ to obtain $-10$.
3. **Add to the next coefficient:**
Add $-10$ to the second coefficient $10$, resulting in $0$.
4. **Multiply again by $-5$:**
Multiply the result $0$ by $-5$ to get $0$.
5. **Add to the last coefficient:**
Add $0$ to the third coefficient $1$, resulting in a remainder of $1$.
The bottom row now shows the quotient coefficients and the remainder:
- **Quotient coefficients:** $[2, 0]$, which represents the quotient polynomial $2x + 0$ (simply $2x$).
- **Remainder:** $1$.
By the polynomial division algorithm, we have:
$$
\text{Dividend} = (\text{Divisor})(\text{Quotient}) + \text{Remainder}.
$$
Substitute the known values:
$$
\text{Dividend} = (x + 5)(2x) + 1.
$$
Now, expand the product:
$$
(x + 5)(2x) = 2x^2 + 10x.
$$
Finally, adding the remainder:
$$
2x^2 + 10x + 1.
$$
Thus, the dividend polynomial is:
$$
\boxed{2x^2 + 10x + 1}.
$$
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