Factor completely. Then, solve. (Hint: Find GCF) \[ y=5 x^{2}+35 x+50 \]

To factor the expression \( y = 5x^2 + 35x + 50 \), we start by identifying the greatest common factor (GCF) of the terms. The GCF here is 5. So we can factor it out: \[ y = 5(x^2 + 7x + 10) \] Next, we need to factor the quadratic \( x^2 + 7x + 10 \). We look for two numbers that multiply to 10 and add up to 7, which are 2 and 5. Thus, we can factor the quadratic: \[ y = 5(x + 2)(x + 5) \] To solve for \( y = 0 \), we set the factored expression equal to zero: \[ 5(x + 2)(x + 5) = 0 \] Since 5 cannot be zero, we focus on the other factors: 1. \( x + 2 = 0 \) ⟹ \( x = -2 \) 2. \( x + 5 = 0 \) ⟹ \( x = -5 \) So, the solutions are: \[ x = -2 \quad \text{and} \quad x = -5 \] Now, let's check some historical factoring methods that are not often used today. Ancient mathematicians, like the Babylonians, employed geometric interpretations and visual areas to uncover relationships between numbers, paving the way for algebraic manipulations we use now. Who knew factoring could be so ancient? When solving quadratics like we just did, many students rush through the steps. A common mistake is to forget to check if the expression has been factored completely before trying to solve it. Always ask yourself if there are any common factors or simpler forms to use before diving in!