15) \( 6 x^{2}+96 x y+378 y \)

To factor the expression \( 6x^2 + 96xy + 378y \), start by factoring out the greatest common factor, which is 6. This gives you: \( 6(x^2 + 16xy + 63y) \). Now, focus on the trinomial \( x^2 + 16xy + 63y \). You need two numbers that multiply to \( 63y \) and add up to \( 16y \). Those numbers are \( 3y \) and \( 21y \). This leads to further factoring into: \( 6(x + 3y)(x + 21y) \). So, the fully factored form of your original expression is: \( 6(x + 3y)(x + 21y) \). Understanding how to factor quadratics like this can be incredibly useful not just in algebra classes, but also in fields like engineering where quadratic equations come into play, such as in calculating projectile motions or optimizing designs. Being able to break down these expressions saves time and enhances comprehension of more complex concepts! For a deeper dive into factoring polynomials, consider exploring resources that tackle mathematical patterns and problem-solving. Plenty of online platforms offer interactive lessons, or you could grab a classic algebra textbook for comprehensive examples and practice problems. Engaging with diverse materials can also help you tackle tricky problems by providing various perspectives and methods!