To solve the first equation, we can start by simplifying both sides:

√(x^2 - 6x + 9) = √4 + 2√3

√(x - 3)^2 = 2 + 2√3

x - 3 = 2 + 2√3

Next, we can isolate x by moving the constant terms to the other side:

x = 2 + 2√3 + 3

x = 5 + 2√3

Therefore, the solution to the first equation is x = 5 + 2√3.

To solve the second equation, we can start by simplifying both sides:

√(x + 2)√(x + 1) = 1 + √2

Next, we can square both sides to eliminate the square roots:

(x + 2)(x + 1) = (1 + √2)^2

x^2 + 3x + 2 = 1 + 2√2 + 2

x^2 + 3x + 1 = 2√2

Next, we can isolate the square root term by moving the constant terms to the other side:

x^2 + 3x - 2√2 = -1

Finally, we can solve for x by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 3, and c = -2√2. Plugging these values into the quadratic formula, we get:

x = (-3 ± √(3^2 - 4(1)(-2√2))) / (2(1))

x = (-3 ± √(9 + 8√2)) / 2

Therefore, the solutions to the second equation are:

x = (-3 + √(9 + 8√2)) / 2

x = (-3 - √(9 + 8√2)) / 2