What is the product? \( \begin{array}{l}(2 x-1)(x+4) \\ 2 x^{2}-4 \\ 2 x^{2}+4 \\ 2 x^{2}+7 x-4 \\ 2 x^{2}-7 x-4\end{array} \)

To find the product, we can expand \( (2x - 1)(x + 4) \). Using the distributive property, we multiply each term in the first binomial by each term in the second: \[ (2x)(x) + (2x)(4) + (-1)(x) + (-1)(4) = 2x^2 + 8x - x - 4 \] Combining like terms gives: \[ 2x^2 + 7x - 4 \] So the product \( (2x - 1)(x + 4) \) is equal to \( 2x^2 + 7x - 4 \). Now, let's dive into some interesting snippets! Did you know that the distributive property has roots that trace back to ancient mathematics? It’s been used since at least ancient Egypt, where they utilized a method called “duplation.” This technique allowed them to double quantities, laying the groundwork for the methods we still use in algebra today! In real-world applications, the expansion of polynomials like \( (2x - 1)(x + 4) \) helps in various fields such as engineering and physics, where it’s crucial to model relationships. For instance, if you were calculating the total cost of producing a certain number of items with fixed costs and varying materials, using such polynomial products can lead you to a clear formula to identify your expenses or profits with ease!