The given expression is:

\[
\frac{-2x^4 - x^2 + 3}{2x + 3}
\]

We can simplify this using polynomial division. Let's perform the long division of \( -2x^4 - x^2 + 3 \) by \( 2x + 3 \).

### Step-by-step polynomial division:

1. Divide the leading term of the numerator by the leading term of the denominator:
   \[
   \frac{-2x^4}{2x} = -x^3
   \]
   Multiply \( -x^3 \) by \( 2x + 3 \), and subtract from the original expression:
   \[
   (-2x^4 - x^2 + 3) - (-x^3 \cdot (2x + 3)) = (-2x^4 - x^2 + 3) - (-2x^4 - 3x^3)
   \]
   Simplify:
   \[
   3x^3 - x^2 + 3
   \]

2. Divide the leading term of the new expression by the leading term of the denominator:
   \[
   \frac{3x^3}{2x} = \frac{3}{2}x^2
   \]
   Multiply \( \frac{3}{2}x^2 \) by \( 2x + 3 \), and subtract:
   \[
   (3x^3 - x^2 + 3) - \left( \frac{3}{2}x^2 \cdot (2x + 3) \right)
   \]
   Simplify the terms, and continue this process.

Let me calculate the rest for you.

The result of dividing \( -2x^4 - x^2 + 3 \) by \( 2x + 3 \) is:

\[
- x^3 + \frac{3}{2}x^2 - \frac{11}{4}x + \frac{33}{8}
\]
with a remainder of:
\[
- \frac{75}{8}
\]

So the full expression is:

\[
- x^3 + \frac{3}{2}x^2 - \frac{11}{4}x + \frac{33}{8} - \frac{75}{8(2x + 3)}
\]

Would you like further details on the process or have any questions?

Here are some related questions:
1. How do you perform synthetic division?
2. What are the steps to find the remainder in polynomial division?
3. Can long division be applied to any polynomial?
4. How do you verify the quotient from division?
5. What is the significance of the remainder in polynomial division?

**Tip:** In polynomial division, the remainder should always have a degree lower than the divisor.

Learn how to solve the polynomial division of -2x⁴ - x² + 3 by 2x + 3 using long division. The detailed step-by-step solution explains how to divide the terms and simplify the expression. Suitable for high school students in Grades 10-12.

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