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I see that you've uploaded an image containing math exercises in French. Here's a summary of the problems:

### **EXERCISE 1**:
1. **Calculations:**
   \[
   \text{(a)} \ (3\sqrt{3} + 5)^2 \quad \text{and} \quad (5 - 3\sqrt{3})^2
   \]
2. **Given the real numbers:**
   \[
   a = \sqrt{52 + 30\sqrt{3}} \quad \text{and} \quad b = \sqrt{52 - 30\sqrt{3}}
   \]
   a) Simplify \(a\) and \(b\).  
   b) Compare \(a\) and \(b\).  
   c) Rewrite without a radical in the denominator for \(\frac{a}{b}\).

3. **Using the values:**
   \[
   X = \sqrt{4\sqrt{3} + 7} \quad \text{and} \quad Y = \sqrt{7 - 4\sqrt{3}}
   \]
   a) Calculate \(X \cdot Y\), \(X^2 + Y^2\), \((X + Y)^2\), and \((X - Y)^2\).  
   b) Calculate \((2 + \sqrt{3})^2\) and \((2 - \sqrt{3})^2\), then give simplified expressions for \(X\) and \(Y\).

### **EXERCISE 2**:
1. **Given the expressions:**
   \[
   P(x) = (2x - 5)^2 - (3x + 2)^2 \quad \text{and} \quad R(x) = (3x - 1)(x + 7) - (2x + 5)(6x - 2) + 1 - 9x^2
   \]
   a) Expand and simplify \(P(x)\) and \(R(x)\) by reducing powers of \(x\).

2. **Additional exercises:**
   a) Show that \(\frac{1}{1 + \sqrt{2}} = \sqrt{2} - 1\).  
   b) Prove that \(R\left(\frac{1}{1 + \sqrt{2}}\right) = -44 + 30\sqrt{2}\).

3. **Factorization:**
   Express \(P(x)\) and \(R(x)\) as products of factors.

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Here are 5 follow-up questions related to the exercises:
1. How do we expand and simplify binomials like \((3\sqrt{3} + 5)^2\)?
2. What is the method for rationalizing a denominator, and how can it be applied to \(\frac{a}{b}\)?
3. What is the approach for simplifying nested square roots, such as in \(X = \sqrt{4\sqrt{3} + 7}\)?
4. How do we apply the difference of squares to \(P(x)\)?
5. What is the strategy for solving quadratic expressions such as those in \(P(x)\) and \(R(x)\)?

**Tip:** When simplifying radical expressions, always check if the terms inside the radical can be factored into perfect squares to simplify the calculation.

Solve the math exercises shown in the uploaded image, including simplification of radicals, calculation of binomials, and solving algebraic expressions.

This page provides solutions for algebraic problems including the simplification of radical expressions, binomial expansions, and factorization. Ideal for students in Grades 9-12, it covers key topics like rationalizing denominators and applying the difference of squares theorem.

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