Simplify the Radical Fraction: √(2x²+4xy+2y²+(x+y)²)/(x+y)

To solve this problem, let's go through each step in detail.

Firstly, consider the expression inside the square root. We need to work with:

$2x^2 + 4xy + 2y^2 + (x+y)^2$

Start by expanding $(x+y)^2$, which is:

$(x+y)^2 = x^2 + 2xy + y^2$

Insert this back into the expression:

$2x^2 + 4xy + 2y^2 + x^2 + 2xy + y^2$

Now combine like terms:

• The $x^2$ terms add up to $3x^2$.
• The $y^2$ terms add up to $3y^2$.
• The $xy$ terms add up to $6xy$.

The expression becomes:

$3x^2 + 6xy + 3y^2$

Notice that this can be factored as a perfect square:

$3(x^2 + 2xy + y^2)$

Recognize that $x^2 + 2xy + y^2$ is $(x+y)^2$, so:

$3(x+y)^2$

Take the square root of the expression:

$\sqrt{3(x+y)^2} = \sqrt{3} \cdot (x+y)$

The original expression under the square root now simplifies, and dividing by $(x+y)$:

$\frac{\sqrt{3} \cdot (x+y)}{(x+y)}$

Cancel the common factor $(x+y)$ from numerator and denominator, leaving:

$\sqrt{3}$

Provided $x+y \neq 0$, the simplified value of the original expression is:

Therefore, the solution to the problem is $\sqrt{3}$.