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The given problem requires performing polynomial long division to divide:

\[
\left(-2x^2 - 7x + 1\right) \div \left(x^2 + 3\right)
\]

We are asked to express the polynomial \( p(x) = -2x^2 - 7x + 1 \) in the form:

\[
p(x) = d(x)q(x) + r(x)
\]

Where:
- \( d(x) \) is the divisor polynomial,
- \( q(x) \) is the quotient polynomial, and
- \( r(x) \) is the remainder polynomial.

### Step 1: Set up the division

We are dividing:

\[
\frac{-2x^2 - 7x + 1}{x^2 + 3}
\]

### Step 2: Perform polynomial long division

1. **First term of the quotient**:  
   Divide the leading term of the dividend by the leading term of the divisor:
   \[
   \frac{-2x^2}{x^2} = -2
   \]
   So, the first term of the quotient \( q(x) \) is \(-2\).

2. **Multiply** the entire divisor by \(-2\):
   \[
   -2(x^2 + 3) = -2x^2 - 6
   \]

3. **Subtract** the result from the original dividend:
   \[
   (-2x^2 - 7x + 1) - (-2x^2 - 6) = -7x + 7
   \]

### Step 3: Determine the remainder

Now, the remainder is \(-7x + 7\), which has a degree lower than the divisor \(x^2 + 3\), so we stop here.

Thus, the division yields:
- Quotient \( q(x) = -2 \)
- Remainder \( r(x) = -7x + 7 \)

### Step 4: Final answer

The polynomial \( p(x) = -2x^2 - 7x + 1 \) can be written as:

\[
p(x) = (x^2 + 3)(-2) + (-7x + 7)
\]

Let me know if you would like further clarification or additional details.

Here are five related questions to explore:
1. How would you perform polynomial division when the divisor has a higher degree than the dividend?
2. How is synthetic division different from polynomial long division?
3. What happens if the remainder is zero in polynomial division?
4. How can polynomial long division help in finding the roots of a polynomial?
5. Can you use polynomial division to simplify rational functions?

**Tip**: When performing polynomial division, always ensure that the terms of both the dividend and divisor are written in descending order of powers of \(x\), and don't forget to account for missing powers by including zero terms.

Use polynomial long division to perform the indicated division: (-2x^2 - 7x + 1) ÷ (x^2 + 3). Then write the polynomial p(x) = -2x^2 - 7x + 1 in the form d(x)q(x) + r(x), where d(x) is the divisor polynomial, q(x) is the quotient polynomial, and r(x) is the remainder polynomial.

Learn how to divide polynomials using long division, specifically (-2x^2 - 7x + 1) divided by (x^2 + 3). This detailed step-by-step solution covers core algebraic concepts, including the polynomial division algorithm, quotient, and remainder.

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