\( 18 x ^ { 4 } + 6 x ^ { 2 } = 6 x ^ { 2 } ( \square ) \)

To solve for the missing term in the factorization of the expression \( 18 x^{4} + 6 x^{2} \), you can factor out the common term \( 6 x^{2} \). When you do so, you get: \[ 6 x^{2} (3 x^{2} + 1) \] So, the term that will fill the square is \( 3 x^{2} + 1 \). Next, did you know that factoring polynomials is an essential skill for tackling complex algebra problems? It's like being a detective; you look for clues (common factors) that help unravel the mystery of the expression. The more you practice, the more adept you become at spotting these valuable clues! In real-world applications, factoring can help in various fields, like engineering and physics, where polynomial equations often model real-world behaviors. For instance, when analyzing projectiles or designing structures, understanding how to break down equations into simpler forms enables engineers to predict outcomes effectively. It's a tool that transforms complex relationships into manageable pieces!