# How do you simplify #(2x+7)/(x^2-y^2)+(-5)/(x^2-2xy+y^2)#?

see below

firstly factorise the denominators

we need the denominators the same

now simplify

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To simplify the expression (2x+7)/(x^2-y^2)+(-5)/(x^2-2xy+y^2), we need to find a common denominator for the two fractions. The common denominator is (x^2-y^2)(x^2-2xy+y^2).

Next, we can rewrite the fractions with the common denominator:

(2x+7)/(x^2-y^2) = (2x+7)(x^2-2xy+y^2)/[(x^2-y^2)(x^2-2xy+y^2)]

(-5)/(x^2-2xy+y^2) = -5(x^2-y^2)/[(x^2-y^2)(x^2-2xy+y^2)]

Now, we can combine the fractions:

(2x+7)(x^2-2xy+y^2)/[(x^2-y^2)(x^2-2xy+y^2)] - 5(x^2-y^2)/[(x^2-y^2)(x^2-2xy+y^2)]

Simplifying further:

(2x^3 - 4x^2y + 2xy^2 + 7x^2 - 14xy + 7y^2 - 5x^2 + 5y^2)/[(x^2-y^2)(x^2-2xy+y^2)]

Combining like terms:

(2x^3 - 4x^2y + 2xy^2 + 7x^2 - 5x^2 - 14xy + 7y^2 + 5y^2)/[(x^2-y^2)(x^2-2xy+y^2)]

Simplifying further:

(2x^3 + 3x^2 - 14xy + 12y^2)/[(x^2-y^2)(x^2-2xy+y^2)]

Therefore, the simplified expression is (2x^3 + 3x^2 - 14xy + 12y^2)/[(x^2-y^2)(x^2-2xy+y^2)].

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