Fsctorize: \( y^{2}+2 a+a y+2 y \) \( \begin{array}{ll}\text { A. }(y-2 a)(y+1) & \text { 8. }(y+2 a)(y-1) \\ \text { C. }\left(y^{2}-1\right)(y-a) & \text { D. }(y+2)(y+a)\end{array} \)

To factor the expression \( y^{2} + 2a + ay + 2y \), we first rearrange terms to make it easier to analyze: \( y^{2} + (a + 2)y + 2a \). Now, we can look for two numbers that multiply to \( 2a \) and add up to \( a + 2 \). These numbers are \( (y + 2)(y + a) \). So, the full factorization is provided in option D: \( (y + 2)(y + a) \). 🎉 Here’s a fun little fact for you: the origins of factoring go back to ancient Babylon, where mathematicians laid the groundwork for algebra without even using symbols the way we do today! They were problem-solving wizards using clay tablets instead! 💡 Remember, when you’re tackling factoring problems, it helps to group terms and look for common factors or patterns! Many students mistakenly jump into complex methods right away, forgetting that sometimes simple rearranging leads to the simplest solution!