fallorise \( 5 x^{2}+11 x+2 \)

To factor the quadratic \( 5x^2 + 11x + 2 \), we look for two numbers that multiply to \( 5 \times 2 = 10 \) and add up to \( 11 \). Those numbers are \( 10 \) and \( 1 \). We can rewrite the middle term, using these numbers, giving us \( 5x^2 + 10x + x + 2 \). Now, we group the first two terms and the last two terms: \( (5x^2 + 10x) + (x + 2) \). Factoring each group gives us \( 5x(x + 2) + 1(x + 2) \). We can factor out the common factor \( (x + 2) \), resulting in \( (5x + 1)(x + 2) \). So the fully factored form is \( (5x + 1)(x + 2) \). --- Did you know that quadratics can represent real-world scenarios, such as projectile motion? By factoring quadratics, you can determine key points like maximum height and landing time, typically essential for engineers and architects. Also, when factoring, a common mistake is overlooking the significance of the leading coefficient (the \( 5 \) in this case), which can complicate finding the right pairs of numbers. Always ensure to take that into account to make your factoring easier!