18.) Factor by grouping. a. \( x^{2}-3 x+2 x-6 \)

To factor by grouping, we start by rearranging the terms: \( (x^2 - 3x) + (2x - 6) \). Next, we can factor out common elements from each group: from the first group, factoring out \( x \) gives us \( x(x - 3) \) and from the second group, factoring out 2 gives us \( 2(x - 3) \). Now we have \( x(x - 3) + 2(x - 3) \). We can factor out the common binomial \( (x - 3) \), leading us to our final factored form: \( (x - 3)(x + 2) \). This method isn’t just useful for solving polynomial equations; it’s a great tool in simplifying expressions in calculus and algebra as well. When you encounter complex polynomials, grouping helps untangle terms systematically, making it easier to derive subsequent derivatives or integrals. Plus, mastering this technique can really boost your algebraic skills and confidence! Don’t be afraid to use it whenever you see polynomial expressions with four or more terms – it can save you a lot of time and trouble!