Use synthetic division to divide the polynomials. $$ \left(-3+4 x+4 x^{3}-2 x^{4}\right) \div(x+2)=\square $$

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Given the division
$$
\frac{-3+4x+4x^3-2x^4}{x+2},
$$
we first rewrite the polynomial in descending order and include any missing degrees. The dividend becomes
$$
-2x^4+4x^3+0x^2+4x-3.
$$

We will use synthetic division with the zero of the divisor $$x+2$$, which is $$x=-2$$.

The coefficients of the dividend are:

$$
-2,\quad 4,\quad 0,\quad 4,\quad -3.
$$

Now, perform synthetic division with $$ -2 $$:

1. Write the coefficients:
$$
\begin{array}{cccccc}
-2 & \quad 4 & \quad 0 & \quad 4 & \quad -3 
\end{array}
$$

2. Bring down the first coefficient:
$$
-2
$$

3. Multiply $$ -2 $$ (the number just written) by $$ -2 $$ (the synthetic divisor) and write the result under the next coefficient:
$$
-2 \times (-2) = 4.
$$

4. Add this to the second coefficient:
$$
4+4=8.
$$

5. Multiply the new number $$8$$ by $$-2$$:
$$
8 \times (-2)=-16.
$$

6. Add this to the third coefficient:
$$
0+(-16)=-16.
$$

7. Multiply $$-16$$ by $$-2$$:
$$
-16 \times (-2)=32.
$$

8. Add this to the fourth coefficient:
$$
4+32=36.
$$

9. Multiply $$36$$ by $$-2$$:
$$
36 \times (-2)=-72.
$$

10. Add this to the fifth coefficient:
$$
-3+(-72)=-75.
$$

The synthetic division table is now complete. The resulting numbers are:

- Quotient coefficients: $$-2$$, $$8$$, $$-16$$, $$36$$ which correspond to the polynomial
$$
-2x^3+8x^2-16x+36.
$$
- Remainder: $$-75$$.

Thus, the division can be written as:
$$
\frac{-2x^4+4x^3+0x^2+4x-3}{x+2} = -2x^3+8x^2-16x+36 + \frac{-75}{x+2}.
$$

The answer (the quotient) is:
$$
\boxed{-2x^3+8x^2-16x+36 \quad \text{with a remainder of} \quad -75.}
$$
The division of $$-2x^4 + 4x^3 + 0x^2 + 4x - 3$$ by $$x + 2$$ using synthetic division yields a quotient of $$-2x^3 + 8x^2 - 16x + 36$$ and a remainder of $$-75$$.

Use synthetic division to divide the polynomials.

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