Question

(x^2+4y^2)/(x-2y)+4xy/(2y-x)

Answer 1

(x²+4y²)/(x-2y) +4xy/(2y-x)
x²+4y²/x-2y+4xy/-(x-2y)
x²+4y²-4xy/x-2y
x²-4xy+4y²/x-2y
(x-2y)²/x-2y
x-2y

Answer 2

To simplify the expression (x^2 + 4y^2)/(x - 2y) + 4xy/(2y - x), we first need to find a common denominator for the two fractions. The common denominator is (x - 2y)(2y - x) since it contains both denominators.

Now, let's proceed with the simplification step by step:

Step 1: Factor the numerator of the first fraction (x^2 + 4y^2).

The numerator (x^2 + 4y^2) is already factored.

Step 2: Find the common denominator.

Common denominator = (x - 2y)(2y - x)

Step 3: Rewrite each fraction with the common denominator.

(x^2 + 4y^2)/(x - 2y) + 4xy/(2y - x) = [(x^2 + 4y^2) * (2y - x)]/[(x - 2y)(2y - x)] + [(4xy) * (x - 2y)]/[(x - 2y)(2y - x)]

Step 4: Combine the fractions.

[(x^2 + 4y^2) * (2y - x) + 4xy * (x - 2y)] / [(x - 2y)(2y - x)]

Step 5: Expand the numerators.

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

Step 6: Combine like terms in the numerator.

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

Step 7: Simplify the expression in the numerator if possible.

[6xy^2 - x^3 - 4y^3] / [(x - 2y)(2y - x)]

That's the simplified expression: (6xy^2 - x^3 - 4y^3) / [(x - 2y)(2y - x)]