Tính: (8 + 2√2)/(3 - √2) - (2 + 3√2)/(√2 + √2)/(1 - √2)

To simplify this expression, we need to rationalize the denominators.

First, let's rationalize the denominator of the first fraction, (8 + 2sqrt(2))/(3 - sqrt(2)):

Multiply the numerator and denominator by the conjugate of the denominator, which is (3 + sqrt(2)):

[(8 + 2sqrt(2))/(3 - sqrt(2))] * [(3 + sqrt(2))/(3 + sqrt(2))] = ([(8 + 2sqrt(2))(3 + sqrt(2)]) / [(3 - sqrt(2))(3 + sqrt(2))])

Expanding the numerator and denominator:

= [(24 + 8sqrt(2) + 6sqrt(2) + 2 * sqrt(2)^2) / (9 - 2)] = [(24 + 14sqrt(2) + 2 * 2) / 7]

= (28 + 14sqrt(2)) / 7 = 4 + 2sqrt(2)

Now, let's rationalize the denominator of the second fraction, (2 + 3sqrt(2))/(sqrt(2)):

Multiply the numerator and denominator by sqrt(2):

[(2 + 3sqrt(2))/(sqrt(2))] * [(sqrt(2))/(sqrt(2))] = [(2sqrt(2) + 3sqrt(2) * sqrt(2)) / (sqrt(2) * sqrt(2))]

Simplifying:

= [(2sqrt(2) + 3sqrt(2)^2) / 2] = [(2sqrt(2) + 3 * 2) / 2]

= (2sqrt(2) + 6) / 2 = sqrt(2) + 3

Finally, let's rationalize the denominator of the third fraction, (sqrt(2))/(1 - sqrt(2)):

Multiply the numerator and denominator by the conjugate of the denominator, which is (1 + sqrt(2)):

[(sqrt(2))/(1 - sqrt(2))] * [(1 + sqrt(2))/(1 + sqrt(2))] = [(sqrt(2))(1 + sqrt(2)) / (1 - sqrt(2))(1 + sqrt(2))]

Expanding the numerator and denominator:

= [(sqrt(2) + sqrt(2)^2) / (1 - sqrt(2) + sqrt(2) - sqrt(2)^2)]

= [(sqrt(2) + 2) / (1 - 2)]

= (sqrt(2) + 2) / -1 = -sqrt(2) - 2

Now, we can substitute these simplified fractions back into the original expression:

(4 + 2sqrt(2)) - (sqrt(2) + 3) + (-sqrt(2) - 2)

Combining like terms:

= 4 + 2sqrt(2) - sqrt(2) - 3 - sqrt(2) - 2

= 4 - 3 - 2 + 2sqrt(2) - sqrt(2) - sqrt(2)

= -1 - sqrt(2)