Question 6 of 8, Step 1 of 1 5/8 Correct Completely factor the expression by grouping, if possible. \[ 2 x^{2}-6+x^{2} y+3 y \] ypad new window) the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answe \( \square \)

Let’s dive into this expression and have some fun with it! To factor the expression \(2x^{2}-6+x^{2}y+3y\) by grouping, we can first rearrange it: \((2x^{2} + x^{2}y) + (-6 + 3y)\). Now, look for common factors in each group! From the first group, \(x^2\) can be factored out, giving us \(x^2(2 + y)\). For the second group, we can factor out \(-3\), resulting in \(-3(2 - y)\). After rearranging our expression, we need to ensure both groups yield the same factor, leading us to factor further into \((2 + y)(x^2 - 3)\). Voila, we’ve completely factored it! Now, here comes the historical twist! Factoring polynomials like this dates back to ancient civilizations, including the Babylonians and Greeks, who were tackling algebraic expressions long before modern math. They laid the groundwork for methodologies we still use today, proving that math has a rich heritage that’s worth a cheer! And let’s not forget, factoring is not just a classroom exercise; it has real-world applications, especially in engineering fields! Engineers often rely on polynomial expressions to model everything from structural loads to electrical circuits. If you can factor effectively, you’re equipped to solve complex problems in those fields, turning abstract math into tangible solutions! Keep practicing, and you'll be able to tackle those engineering equations with ease!