(x/(x^2-8x+16)-x+6/(x^2-16)):x+12/x^2-16

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вопрос

To simplify the given expression, we first need to factor the denominators in each term:

x^2 - 8x + 16 = (x - 4)^2
x^2 - 16 = (x - 4)(x + 4)

Now we rewrite the expression with factored denominators:

x/((x - 4)^2) - x + 6/((x - 4)(x + 4)) : x + 12/((x - 4)(x + 4))

Next, we find a common denominator for all the terms in the expression, which is (x - 4)^2(x + 4):

[(x(x + 4))/((x - 4)^2) - x((x - 4)(x + 4))/((x - 4)^2(x + 4)) + 6(x - 4)^2/((x - 4)^2(x + 4))] : [x(x - 4)^2(x + 4)/((x - 4)^2(x + 4)) + 12((x - 4)(x + 4))/((x - 4)^2(x + 4))]

Now simplify the expression by cancelling out common factors:

[(x(x + 4) - x(x - 4)(x + 4) + 6(x - 4)^2)]: [(x(x - 4)^2(x + 4) + 12(x - 4)(x + 4))]

[(x^2 + 4x - x(x^2 + 4x - 4x - 16) + 6(x^2 - 8x + 16))]: [(x(x^2 - 4x + 4)(x + 4) + 12(x^2 - 4x - 4x + 16))]

[(x^2 + 4x - x^3 - 4x^2 + 4x + 16 + 6x^2 - 48x + 96)]: [(x^3 - 4x^2 + 4x)(x + 4) + 12x^2 - 48x + 48]

[-x^3 + 7x^2 - 44x + 112]: [x^4 + 4x^2 - 16x + 12x^2 - 48x + 48]

Finally, the expression simplifies to:

(-x^3 + 7x^2 - 44x + 112) / (x^4 + 16x^2 - 64x + 48)