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Simplify 3/x-15/2x^2+5x + 2/2x+5

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Answer: To simplify the expression, let's find a common denominator and combine the terms:

The common denominator for the fractions is (2x^2 + 5x)(2x + 5).

Express each fraction with the common denominator:

3/x = (3 * 2x + 3 * 5) / (x * (2x + 5))

= (6x + 15) / (x * (2x + 5))

-15/ (2x^2 + 5x) = (-15) / ((2x^2 + 5x) * (2x + 5))

2/(2x + 5) = (2 * x) / ((2x + 5) * x)

= 2x / (2x^2 + 5x)

Now, the simplified expression becomes:

(6x + 15) / (x * (2x + 5)) - (15) / ((2x^2 + 5x) * (2x + 5)) + 2x / (2x^2 + 5x)

To combine the fractions, find a common denominator for all terms, which is (x * (2x + 5)) * ((2x^2 + 5x) * (2x + 5)).

Express each fraction with the common denominator:

[(6x + 15) * ((2x^2 + 5x) * (2x + 5)) - 15 * (x * (2x + 5)) + 2x * (x * (2x + 5))] / (x * (2x + 5)) * ((2x^2 + 5x) * (2x + 5))

Simplify the numerator:

(6x + 15) * (2x^3 + 5x^2 + 10x + 25) - 15 * (2x^2 + 5x) + 2x * (2x^2 + 5x)

Expand and simplify:

(12x^4 + 30x^3 + 60x^2 + 150x + 30x^3 + 75x^2 + 150x + 375) - (30x^2 + 75x) + 4x^3 + 10x^2

Combine like terms:

12x^4 + 34x^3 + 85x^2 + 285x + 375 - 30x^2 - 75x + 4x^3 + 10x^2

Simplify further:

12x^4 + 34x^3 - 25x^2 + 210x + 375

So, the simplified expression is 12x^4 + 34x^3 - 25x^2 + 210x + 375.

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