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The problem asks to perform polynomial long division to divide:

\[
12x^3 - 7x^2 + 24x + 26 \text{ by } 4x + 3.
\]

We are tasked with expressing the result in the form:
\[
q(x) + \frac{r(x)}{b(x)},
\]
where \( q(x) \) is the quotient, \( r(x) \) is the remainder, and \( b(x) \) is the divisor \( 4x + 3 \).

### Steps for Polynomial Long Division:

1. **Divide the first term**: Divide \( 12x^3 \) by \( 4x \), which gives \( 3x^2 \).
2. **Multiply and subtract**: Multiply \( 3x^2 \) by \( 4x + 3 \), which gives:
   \[
   3x^2 \cdot (4x + 3) = 12x^3 + 9x^2.
   \]
   Subtract this from the original polynomial:
   \[
   (12x^3 - 7x^2) - (12x^3 + 9x^2) = -16x^2.
   \]
3. **Repeat**: Divide \( -16x^2 \) by \( 4x \), which gives \( -4x \). Multiply and subtract:
   \[
   -4x \cdot (4x + 3) = -16x^2 - 12x.
   \]
   Subtract from the new polynomial:
   \[
   (-16x^2 + 24x) - (-16x^2 - 12x) = 36x.
   \]
4. **Continue**: Divide \( 36x \) by \( 4x \), which gives \( 9 \). Multiply and subtract:
   \[
   9 \cdot (4x + 3) = 36x + 27.
   \]
   Subtract:
   \[
   (36x + 26) - (36x + 27) = -1.
   \]
5. **Conclusion**: The remainder is \( -1 \), so the quotient \( q(x) = 3x^2 - 4x + 9 \) and the remainder \( r(x) = -1 \).

### Final Answer:
The result of the division is:

\[
3x^2 - 4x + 9 + \frac{-1}{4x + 3}.
\]

Let me know if you need any further clarification!

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**Relative questions:**
1. How would the result change if the divisor were \( 2x + 1 \)?
2. What is the general method to check if a remainder exists?
3. Can synthetic division be used here, and why or why not?
4. What are the implications of having a remainder in polynomial division?
5. How does the degree of the remainder relate to the degree of the divisor?

**Tip:** Always make sure the degree of the remainder is less than the degree of the divisor when performing polynomial division.

Use the long division method to find the result when 12x^3 - 7x^2 + 24x + 26 is divided by 4x + 3. If there is a remainder, express the result in the form q(x) + r(x) / b(x).

Learn how to use polynomial long division to divide 12x^3 - 7x^2 + 24x + 26 by 4x + 3. Find the quotient and remainder, and express the result in the form q(x) + r(x) / b(x). This detailed solution covers key algebraic concepts suitable for students in Grades 10-12.

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