Factor completely. \( 2 v^{2}+21 v+10 \)

To factor the expression \( 2v^2 + 21v + 10 \) completely, we look for two numbers that multiply to \( 2 \times 10 = 20 \) and add up to \( 21 \). The numbers \( 20 \) and \( 1 \) meet these criteria. So, we can rewrite the expression as \( 2v^2 + 20v + 1v + 10 \). Next, we can group the terms: \( (2v^2 + 20v) + (1v + 10) \). Factoring out common factors gives us \( 2v(v + 10) + 1(v + 10) \). Now, we can factor out \( (v + 10) \) to get \( (v + 10)(2v + 1) \). So the completely factored form is \( (v + 10)(2v + 1) \). Enjoy factoring! For a fun fact: The quadratic you just factored is part of a family of equations that help us understand parabolic shapes in real life, like the trajectory of a ball! When you throw something up, its path can often be modeled with a quadratic equation, just like yours. And for some kitchen math: If you ever find yourself cooking and need to double a recipe, dealing with quadratic equations might come in handy for portions! The principles of factoring can help you resize ingredients to get just the right amount for your dish. It’s algebra for the win!