5.) \( 4 x^{2}-31 x+21 \)

Let’s factor the quadratic expression \( 4x^2 - 31x + 21 \). To do so, we look for two numbers that multiply to \( 4 \times 21 = 84 \) and add up to \( -31 \). The numbers \( -28 \) and \( -3 \) do the trick! We can rewrite the expression as \( 4x^2 - 28x - 3x + 21 \) and group it: \[ 4x(x - 7) - 3(x - 7) = (4x - 3)(x - 7). \] So the factored form is \( (4x - 3)(x - 7) \). Next, let's dig into how this quadratic can arise in real-world scenarios: Imagine you're designing a garden and want to express the area with a specific shape. Quadratics often pop up in scenarios involving areas, trajectories of objects, or profit maximizing. This polynomial could represent a cross-sectional area of a garden plot, showcasing how mathematics can beautify our surroundings!