Fully simplify each of the foll i. \( (2 x+y)\left(3 x^{2}+2 x y+5 y^{2}\right) \)

To simplify the expression \( (2x + y)(3x^{2} + 2xy + 5y^{2}) \), we can use the distributive property, also known as the FOIL method for binomials. 1. Distributing \( 2x \) to each term in the second polynomial: - \( 2x \cdot 3x^{2} = 6x^{3} \) - \( 2x \cdot 2xy = 4x^{2}y \) - \( 2x \cdot 5y^{2} = 10xy^{2} \) 2. Distributing \( y \) to each term in the second polynomial: - \( y \cdot 3x^{2} = 3x^{2}y \) - \( y \cdot 2xy = 2xy^{2} \) - \( y \cdot 5y^{2} = 5y^{3} \) Now, we can combine all the results: \[ 6x^{3} + 4x^{2}y + 10xy^{2} + 3x^{2}y + 2xy^{2} + 5y^{3} \] Next, we combine like terms: - \( 4x^{2}y + 3x^{2}y = 7x^{2}y \) - \( 10xy^{2} + 2xy^{2} = 12xy^{2} \) Thus, the fully simplified expression is: \[ 6x^{3} + 7x^{2}y + 12xy^{2} + 5y^{3} \]