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The expression you provided in the image is a polynomial division setup, where the divisor is \( x + 3 \), and the dividend is \( x^4 + 24x^2 + 3x - 7 \). This represents a long division of polynomials.

Let me walk through the steps of solving this polynomial division.

1. **Step 1: Divide the leading term**:
   Divide the leading term of the dividend, \( x^4 \), by the leading term of the divisor, \( x \):
   \[
   \frac{x^4}{x} = x^3
   \]
   So, the first term of the quotient is \( x^3 \).

2. **Step 2: Multiply and subtract**:
   Multiply \( x^3 \) by \( x + 3 \) to get:
   \[
   x^3(x + 3) = x^4 + 3x^3
   \]
   Now subtract \( (x^4 + 3x^3) \) from \( (x^4 + 24x^2 + 3x - 7) \):
   \[
   (x^4 + 24x^2 + 3x - 7) - (x^4 + 3x^3) = -3x^3 + 24x^2 + 3x - 7
   \]

3. **Step 3: Repeat the process**:
   - Divide the new leading term, \( -3x^3 \), by \( x \):
     \[
     \frac{-3x^3}{x} = -3x^2
     \]
     So, the next term in the quotient is \( -3x^2 \).

   - Multiply \( -3x^2 \) by \( x + 3 \):
     \[
     -3x^2(x + 3) = -3x^3 - 9x^2
     \]
     Subtract this from the remaining polynomial:
     \[
     (-3x^3 + 24x^2 + 3x - 7) - (-3x^3 - 9x^2) = 33x^2 + 3x - 7
     \]

4. **Step 4: Continue dividing**:
   - Divide \( 33x^2 \) by \( x \):
     \[
     \frac{33x^2}{x} = 33x
     \]
     The next term in the quotient is \( 33x \).

   - Multiply \( 33x \) by \( x + 3 \):
     \[
     33x(x + 3) = 33x^2 + 99x
     \]
     Subtract this from the polynomial:
     \[
     (33x^2 + 3x - 7) - (33x^2 + 99x) = -96x - 7
     \]

5. **Step 5: Final division**:
   - Divide \( -96x \) by \( x \):
     \[
     \frac{-96x}{x} = -96
     \]
     The final term in the quotient is \( -96 \).

   - Multiply \( -96 \) by \( x + 3 \):
     \[
     -96(x + 3) = -96x - 288
     \]
     Subtract:
     \[
     (-96x - 7) - (-96x - 288) = 281
     \]

The quotient is \( x^3 - 3x^2 + 33x - 96 \) with a remainder of 281.

### Final Answer:
\[
\frac{x^4 + 24x^2 + 3x - 7}{x + 3} = x^3 - 3x^2 + 33x - 96 + \frac{281}{x + 3}
\]

Would you like further details or have any questions?

Here are some related questions:
1. What is synthetic division, and how is it used in polynomial division?
2. Can this result help in factoring the original polynomial?
3. How would the result change if the divisor were \( x - 3 \) instead of \( x + 3 \)?
4. What is the Remainder Theorem, and how does it apply to this division?
5. How does the degree of the quotient relate to the degrees of the dividend and divisor?

**Tip**: Polynomial division is useful for simplifying expressions and solving polynomial equations where factorization is challenging.

Polynomial division problem: \( \frac{x^4 + 24x^2 + 3x - 7}{x + 3} \).

Explore how to solve the polynomial division \( \frac{x^4 + 24x^2 + 3x - 7}{x + 3} \) using long division. Learn key algebraic concepts and the remainder theorem with a detailed step-by-step guide. This solution is suitable for high school students in Grades 10-12.

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