(3*sqrt(2\3)-sqrt(24)+sqrt(6))(2*sqrt(2\3)+3*sqrt(2\3))

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To simplify this expression, we first need to distribute the terms inside the parentheses:

(3sqrt(2/3) - sqrt(24) + sqrt(6)) (2sqrt(2/3) + 3sqrt(2/3))

First distribute the terms inside the parentheses:

= 6(2/3) + 9(2/3) + 2(2/3)sqrt(2/3) + 3(2/3)sqrt(2/3) - 2sqrt(24) - 3sqrt(24) + 2sqrt(6) - 3sqrt(6)

Next, simplify the products:

= 4 + 6 + 4sqrt(2/3) + 6sqrt(2/3) - 2sqrt(24) - 3sqrt(24) + 2sqrt(6) - 3sqrt(6)

Now we need to simplify the square roots:

sqrt(2/3) = sqrt(2)/sqrt(3) = sqrt(2)/√3 (rationalizing the denominator)

sqrt(24) = sqrt(46) = sqrt(4)sqrt(6) = 2*sqrt(6)

sqrt(6) remains as is

Substitute these values back into the expression:

= 4 + 6 + 4(sqrt(2)/√3) + 6(sqrt(2)/√3) - 22sqrt(6) - 32sqrt(6) + 2sqrt(6) - 3sqrt(6)

= 10 + 10(sqrt(2)/√3) - 4sqrt(6) - 6sqrt(6) + 2sqrt(6) - 3*sqrt(6)

= 10 + 10(sqrt(2)/√3) - 8sqrt(6) - 7*sqrt(6)

= 10 + 10(sqrt(2)/√3) - 15sqrt(6)

Therefore,

(3sqrt(2/3) - sqrt(24) + sqrt(6)) (2sqrt(2/3) + 3sqrt(2/3)) = 10 + 10(sqrt(2)/√3) - 15sqrt(6)