ta có:

a/ khi x = 4 + 2√3, ta có:

P = (√x + 3)/(√x + 2) + (√x + 2)/(√x + 1) - (3√x + 6)/(x + 3√x + 2)

= (√(4 + 2√3) + 3)/(√(4 + 2√3) + 2) + (√(4 + 2√3) + 2)/(√(4 + 2√3) + 1) - (3√(4 + 2√3) + 6)/(4 + 2√3 + 3√(4 + 2√3) + 2)

= (√[(√3 + 1)^2] + 3)/(√[(√3 + 1)^2] + 2) + (√[(√3 + 1)^2] + 2)/(√[(√3 + 1)^2] + 1) - (3√[(√3 + 1)^2] + 6)/(4 + 2√3 + 3√[(√3 + 1)^2] + 2)

= (√3 + 1 + 3)/(√3 + 1 + 2) + (√3 + 1 + 2)/(√3 + 1 + 1) - (3(√3 + 1) + 6)/(4 + 2√3 + 3(√3 + 1) + 2)

= (√3 + 4)/(√3 + 3) + (√3 + 3)/(√3 + 2) - (3√3 + 9)/(9 + 5√3)

= [(√3 + 4)(√3 + 2) + (√3 + 3)(√3 + 3) - (3√3 + 9)] / [(√3 + 3)(√3 + 2)]

= (3 + 6√3 + 8 + 6 + 6√3 + 9 - 3√3 - 9) / (3 + 5√3 + 6)

= (17 + 9√3) / (9 + 5√3)

= [(17 + 9√3)(9 - 5√3)] / [(9 + 5√3)(9 - 5√3)]

= (153 - 85√3 + 81√3 - 135) / (81 - 75)

= (18 - 4√3) / 6

= 3 - 2√3 / 3

vậy khi x = 4 + 2√3 thì P = 3 - 2√3 / 3

b/ ta có:

Q = 1/(√x + 1) + 1/(√x + 2) + 1/(√x + 3) - 1/(√x + 4) - 1/(√x + 5) - 1/(√x + 6)

= [1/(√x + 1) - 1/(√x + 4)] + [1/(√x + 2) - 1/(√x + 5)] + [1/(√x + 3) - 1/(√x + 6)]

= (√x + 4 - √x - 1)/[(√x + 1)(√x + 4)] + (√x + 5 - √x - 2)/[(√x + 2)(√x + 5)] + (√x + 6 - √x - 3)/[(√x + 3)(√x + 6)]

= 3/[(√x + 1)(√x + 4)] + 3/[(√x + 2)(√x + 5)] + 3/[(√x + 3)(√x + 6)]

= 3 * [(√x + 2)(√x + 5)(√x + 6) + (√x + 1)(√x + 4)(√x + 6) + (√x + 1)(√x + 4)(√x + 2)(√x + 5)] / [(√x + 1)(√x + 2)(√x + 3)(√x + 4)(√x + 5)(√x + 6)]

= 3 * [ (x + 7√x + 10)(√x + 6) + (x + 5√x + 4)(√x + 6) + (x + 3√x + 2)(x + 9√x + 20) ] / [(x + 5√x + 4)(x + 5√x + 6)(x + 5√x + 8)]

= 3 * [ x√x + 6x + 7x + 42√x + 10√x + 60 + x√x + 6x + 5x + 30√x + 4√x + 24 + x^2 + 9x√x + 20x + 3x√x + 27x + 60√x + 2x + 18√x + 40 ] / [(x + 5√x + 4)(x^2 + 10x√x + 47x + 110√x + 120)]

= 3 * [ x^2 + 16x√x + 50x + 150√x + 124 ] / [(x + 5√x + 4)(x^2 + 10x√x + 47x + 110√x + 120)]

= 3 * (x + 2)(x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 2)(x + 5)(x + 3)]

= 3 * (x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 5)(x + 3)]

ta có:

P = Q <=> (√x + 3)/(√x + 2) + (√x + 2)/(√x + 1) - (3√x + 6)/(x + 3√x + 2) = 3 * (x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 5)(x + 3)]

<=> [(√x + 3)(√x + 1) + (√x + 2)^2 - (3√x + 6)] / [(√x + 2)(√x + 1)] = 3 * (x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 5)(x + 3)]

<=> (x + 4√x + 3 + x + 4√x + 4 - 3√x - 6) / (x + 3√x + 2) = 3 * (x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 5)(x + 3)]

<=> (2x + 5√x + 1) / (x + 3√x + 2) = 3 * (x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 5)(x + 3)]

<=> (2x + 5√x + 1) / [(√x + 1)(√x + 2)] = 3 * (x + 25 + 8√x) / [(x + 4)(x + 1)(x + 6)(x + 5)(x + 3)]

<=> (2x + 5√x + 1) * (x + 4)(x + 6)(x + 5)(x + 3) = 3 * (x + 25 + 8√x) * (x + 2)

<=> (2x + 5√x + 1) * (x^2 + 10x + 24)(x^2 + 8x + 15) = 3 * (x + 25 + 8√x) * (x + 2)

<=> (2x + 5√x + 1) * (x^4 + 18x^3 + 131x^2 + 450x + 360) = 3 * (x^2 + 27x + 50 + 8x√x + 160√x + 16√x^3)

<=> 2x^5 + 3