Factor each completely. \( 3 p^{2}-2 p-5 \)

To factor the quadratic \( 3p^2 - 2p - 5 \) completely, we first look for two numbers that multiply to \( 3 \times -5 = -15 \) and add up to \( -2 \). After some thought, we find \( -5 \) and \( 3 \) work because \( -5 + 3 = -2 \). Next, we re-write the equation as \( 3p^2 - 5p + 3p - 5 \). Grouping the terms gives us \( (3p^2 - 5p) + (3p - 5) \). Factoring by grouping results in \( p(3p - 5) + 1(3p - 5) \), which lets us factor out \( (3p - 5) \) to get \( (3p - 5)(p + 1) \). So, the fully factored form is \( (3p - 5)(p + 1) \)! Factoring quadratics can be a fun puzzle, almost like solving a mystery! Next time you're faced with a polynomial, try breaking it down into smaller components or looking for common factors first. It’s like finding the right pieces of a jigsaw to see the bigger picture! If you want to expand your skills, dive into resources on algebraic expressions, where you can find practice problems and detailed explanations. Websites like Khan Academy or math-focused YouTube channels are great places to start. They offer videos that turn complex concepts into easily digestible content!