Simplify (x^3+x^2y+xy^2+y^3)/(x^2+2xy+y^2)+(2xy)/(x+y)

$\frac{(x^{3} + x^{2} y + x y^{2} + y^{3})}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Simplify terms.

Simplify each term.

Simplify the numerator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x^{3} + x^{2} y) + x y^{2} + y^{3}}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Factor out the greatest common factor (GCF) from each group.

$\frac{x^{2} (x + y) + y^{2} (x + y)}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Factor the polynomial by factoring out the greatest common factor, $x + y$ .

$\frac{(x + y) (x^{2} + y^{2})}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Factor using the perfect square rule.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Rewrite the polynomial.

$\frac{(x + y) (x^{2} + y^{2})}{x^{2} + 2 \cdot x \cdot y + y^{2}} + \frac{2 x y}{x + y}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = y$ .

$\frac{(x + y) (x^{2} + y^{2})}{{(x + y)}^{2}} + \frac{2 x y}{x + y}$

Cancel the common factors.

Factor $x + y$ out of ${(x + y)}^{2}$ .

$\frac{(x + y) (x^{2} + y^{2})}{(x + y) (x + y)} + \frac{2 x y}{x + y}$

Cancel the common factor.

$\frac{(x + y) (x^{2} + y^{2})}{(x + y) (x + y)} + \frac{2 x y}{x + y}$

Rewrite the expression.

$\frac{x^{2} + y^{2}}{x + y} + \frac{2 x y}{x + y}$

Combine the numerators over the common denominator.

$\frac{x^{2} + y^{2} + 2 x y}{x + y}$

Factor using the perfect square rule.

Rearrange terms.

$\frac{x^{2} + 2 x y + y^{2}}{x + y}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot y + y^{2}}{x + y}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = y$ .

$\frac{{(x + y)}^{2}}{x + y}$

Cancel the common factor of ${(x + y)}^{2}$ and $x + y$ .

Factor $x + y$ out of ${(x + y)}^{2}$ .

$\frac{(x + y) (x + y)}{x + y}$

Cancel the common factors.

Multiply by $1$ .

$\frac{(x + y) (x + y)}{(x + y) \cdot 1}$

Cancel the common factor.

$\frac{(x + y) (x + y)}{(x + y) \cdot 1}$

Rewrite the expression.

$\frac{x + y}{1}$

Divide $x + y$ by $1$ .

$x + y$

Simplify (x^3+x^2y+xy^2+y^3)/(x^2+2xy+y^2)+(2xy)/(x+y)

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