Factor the expression completely. \[ x^{5} y^{2}+y^{5} \]

To factor the expression \( x^{5} y^{2} + y^{5} \), we can first recognize that both terms share a common factor of \( y^2 \). This allows us to rewrite the expression as: \[ y^2 (x^5 + y^3) \] Next, we should analyze the term \( x^5 + y^3 \). This can be seen as a sum of cubes since \( x^5 = (x^{5/3})^3 \) and \( y^3 = (y)^3\). We can apply the sum of cubes factorization: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] In our case, let \( a = x^{5/3} \) and \( b = y \). We can factor \( x^5 + y^3 \) as follows: \[ x^5 + y^3 = (x^{5/3} + y)((x^{5/3})^2 - x^{5/3}y + y^2) = (x^{5/3} + y)(x^{10/3} - x^{5/3}y + y^2) \] Putting everything together, we arrive at the completely factored expression: \[ y^2 (x^{5/3} + y)(x^{10/3} - x^{5/3}y + y^2) \] Thus, the expression \( x^{5} y^{2} + y^{5} \) is completely factored as shown.