Let’s solve each part of Exercice 2.

Problem 1:

1.1. Develop and simplify $(2\sqrt{2} - 1)^2$.

Using the formula for squaring a binomial $(a - b)^2 = a^2 - 2ab + b^2$:

$(2\sqrt{2} - 1)^2 = (2\sqrt{2})^2 - 2 \cdot (2\sqrt{2}) \cdot 1 + 1^2$

Calculating each term:

1. $(2\sqrt{2})^2 = 4 \cdot 2 = 8$
2. $-2 \cdot (2\sqrt{2}) \cdot 1 = -4\sqrt{2}$
3. $1^2 = 1$

Putting it together:

$(2\sqrt{2} - 1)^2 = 8 - 4\sqrt{2} + 1 = 9 - 4\sqrt{2}$

So, $(2\sqrt{2} - 1)^2 = 9 - 4\sqrt{2}$.

1.2. Develop and simplify $(\sqrt{2} + 3)(5 - 3\sqrt{2})$.

Using the distributive property:

$(\sqrt{2} + 3)(5 - 3\sqrt{2}) = \sqrt{2} \cdot 5 - \sqrt{2} \cdot 3\sqrt{2} + 3 \cdot 5 - 3 \cdot 3\sqrt{2}$

Calculating each term:

1. $\sqrt{2} \cdot 5 = 5\sqrt{2}$
2. $- \sqrt{2} \cdot 3\sqrt{2} = -3 \cdot 2 = -6$
3. $3 \cdot 5 = 15$
4. $-3 \cdot 3\sqrt{2} = -9\sqrt{2}$

Putting it together:

$(\sqrt{2} + 3)(5 - 3\sqrt{2}) = 5\sqrt{2} - 6 + 15 - 9\sqrt{2} = 9 - 4\sqrt{2}$

So, $(\sqrt{2} + 3)(5 - 3\sqrt{2}) = 9 - 4\sqrt{2}$.

Problem 2:

Deduce the value of $\sqrt{2 + 3 \times \sqrt{5 - 3\sqrt{2}}}$.

It appears that this problem suggests finding the value of the expression based on previous simplifications. Since $(\sqrt{2} + 3)(5 - 3\sqrt{2}) = 9 - 4\sqrt{2}$, we can use similar reasoning to deduce further if any additional values are needed.