Divide (2x3+4x2+5x−6)÷(x+1) using synthetic division. Enter the missing term in the quotient. (1 point)

To divide the polynomial \(2x^3 + 4x^2 + 5x - 6\) by \(x + 1\) using synthetic division, we first rewrite the divisor in a way that facilitates synthetic division. We can use \(c = -1\) (since \(x + 1 = 0\) leads to \(x = -1\)).

The coefficients of the dividend \(2x^3 + 4x^2 + 5x - 6\) are \(2\), \(4\), \(5\), and \(-6\).

Setting up synthetic division:

1. Write the coefficients: \(2, 4, 5, -6\).
2. Use \(-1\) as the number to perform synthetic division.

Now, perform synthetic division:

```
-1 | 2 4 5 -6
| -2 -2 -3
---------------------
2 2 3 -9
```

Here's what happens step-by-step:

- Bring down the first coefficient \(2\).
- Multiply \(-1\) by \(2\) and write it under the next coefficient (4): \( -1 \cdot 2 = -2\).
- Add \(4 + (-2) = 2\).
- Multiply \(-1\) by \(2\) and write it under the next coefficient (5): \( -1 \cdot 2 = -2\).
- Add \(5 + (-2) = 3\).
- Multiply \(-1\) by \(3\) and write it under the next coefficient (-6): \( -1 \cdot 3 = -3\).
- Add \(-6 + (-3) = -9\).

The synthetic division gives us the remainder \(-9\) and the coefficients of the quotient polynomial as \(2\), \(2\), and \(3\).

The quotient is therefore:

\[
2x^2 + 2x + 3
\]

Since we are asked for a missing term in the quotient, it would be more accurate to note that there is no missing term in the polynomial, but if you were searching for a particular coefficient, \(2, 2, 3\) correspond to \(x^2, x, \) and the constant term respectively.

So the final answer is:

Quotient: \(2x^2 + 2x + 3\)

We have no missing term since all coefficients are accounted for.