A function is defined as \( f(x)=2 x^{2}-x-6 \), where \( x \in \mathbb{R} \). (a) Write down the two factors of \( f(x) \). \[ \begin{array}{lll}2 x^{2}-x-6 & \text { T.U.12 } & (2 x+3)(x-2) \\ 2 x^{2}-4 x+3 x-6 & 2 x+3=0 & x=2 \\ 2 x(x-2) 3(x-2) & 2 x=-3\end{array} \begin{array}{l}12 \times 1 \\ 1 \times 12 \\ 2 \times 6 \\ 6 \times 2 \\ 4 \times 3 \\ 3 \times 4\end{array} \]

To factor the quadratic function \( f(x) = 2x^2 - x - 6 \), we need to express it in the form \( a(x - p)(x - q) \), where \( p \) and \( q \) are the roots. Using the factoring method or the quadratic formula, we find that the factors are \( (2x + 3)(x - 2) \). Did you know that factoring quadratics isn't just a math trick? It’s a powerful technique that has real-world applications, like in engineering and physics, where predicting the behavior of falling objects or optimizing designs relies on understanding quadratic relationships! Also, when factoring, a common mistake is to overlook the need for being systematic. Remember to look for coefficients that add up to the middle term when breaking down the expression. Factor by grouping or using the quadratic formula as a safeguard to avoid errors!