https://file.aiquickdraw.com/mathbot/file/1728814117570e9wvcgxu.jpg

I can help you with the mathematical problems from the image you uploaded. Let's begin with **Partie I**:

### Partie I:
1. **Étudier la parité de l'expression suivante:**
   \[
   A = (6n + 11)(-1)^{n+8} + 8(2n + 3)(-1)^{n+5}
   \]
   - Here, you are asked to study the **parity** (whether the expression is odd or even) based on the value of \( n \). The parity can be analyzed by checking how powers of \(-1\) and multiples of numbers affect the result.

2. **Montrer que le nombre \( (6n + 12)^2 + (2n + 4)^2 \) est un multiple de 40:**
   - Expand the squares and check whether the result is divisible by 40 for all \( n \in \mathbb{N} \).

3. **Soit \( n \geq 2 \), poser \( a = 8n^2 + 6n + 1 \) et \( b = 21n^2 + 2 \):**
   - **(a)** Factor these expressions and simplify.
   - **(b)** Find the **PGCD (gcd)** and **PPCM (lcm)** of \( a \) and \( b \).

### Partie II:
1. **Soit \( p \) un entier naturel impair, montrer que \( 8 \) divise \( p^2 - 1 \):**
   - This requires factoring \( p^2 - 1 \) and proving its divisibility by 8 for odd values of \( p \).

2. **Simplify the following expressions:**
   \[
   A = 2 + \frac{1}{\sqrt{3}}, \quad B = (1 + \sqrt{2})^3, \quad C = \sqrt{2\sqrt{3} + 7} + \sqrt{5}, \quad D = (4\sqrt{3} - 7)^{2024} \times (4\sqrt{3} + 7)^{2024}
   \]
   - Simplify each expression.

3. **Simplify and determine the set of \( E \) and \( F \):**
   \[
   E = \frac{7001}{7002} - \frac{7002}{7003}, \quad F = \frac{4n^2 + 3n + 2}{3} + \frac{1}{2} \times \left(\frac{2^{n+8}}{2n^2 + 2n + 1}\right)
   \]

4. **Factor the following expression:**
   \[
   G = (x\sqrt{x} - 1) - 5(1 - x) - 3(x\sqrt{x} - 1)^2
   \]

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Would you like detailed steps for any of these problems, or do you have a specific part you'd like to start with? 

### Related Questions:
1. How do powers of \( -1 \) affect the parity of an expression?
2. What is the significance of finding the GCD and LCM in algebraic expressions?
3. Can you describe a method for proving divisibility by 8?
4. How can the binomial expansion help in simplifying complex powers?
5. What strategies can be used to factor complex expressions?

### Tip:
When simplifying radical expressions, try rationalizing the denominator or using algebraic identities to make calculations more straightforward.

Math problems involving algebraic expressions, parity, divisibility, and factorization.

Solve problems on parity, divisibility, factorization, and algebraic expressions, including the study of GCD and LCM. Learn to simplify complex expressions and understand the divisibility by numbers like 8 and 40.

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