https://file.aiquickdraw.com/mathbot/file/1725508545902c4lire2h.jpg

Let's go through each question step by step from the image you've uploaded:

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1. **Given**:

   \[
   \frac{a + b}{2x - 3} = \frac{10x + 1}{6x^2 - 11x + 3}
   \]

   You need to find the value of \( a + b \).

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2. **Given**:

   \[
   \frac{2x + 3}{x + 2} = A + \frac{B}{x + C}
   \]

   You need to determine the values of \( A \), \( B \), and \( C \).

---

3. **Find the quotient and remainder for the division**:

   \[
   \frac{6x^2 - 2x - x^2}{x + 2}
   \]

   Perform polynomial division.

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4. **Find the quotient and remainder for the division**:

   \[
   \frac{x^4 + 2x^3 - x^2 - 8x + 3}{x^2 + 3x + 1}
   \]

   Perform polynomial division.

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5. **Find the remainder when the polynomial**:

   \[
   f(x) = 6x^3 + x^2 - 5x + 11
   \]

   is divided by \( x - 1 \).

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6. **Find the value of**:

   \[
   f(-1) \text{ for the polynomial } 4x^4 - 12x^3 + 3x - 7
   \]

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7. **Given the polynomial**:

   \[
   6x^3 + 7x^2 + Px - 24
   \]

   is divisible by \( 2x - 3 \), find the value of \( P \).

---

I can begin solving them one by one. Let me know if you'd like the solution for all or specific ones.

The image contains multiple mathematical questions related to algebra and polynomial functions, including finding the values of variables and solving polynomial equations.

This page provides a detailed solution to algebraic polynomial problems, including polynomial division, finding remainders, and solving rational functions. Learn key concepts like the Remainder Theorem and polynomial factorization. Suitable for high school students in Grades 10-12.

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