Solve by factoring. \[ 5 x^{2}+16 x-19=11-3 x \]

First, let’s rearrange the equation \( 5 x^{2} + 16 x - 19 = 11 - 3 x \) to one side. We move all terms to the left: \[ 5 x^{2} + 16 x - 19 - 11 + 3 x = 0 \] This simplifies to: \[ 5 x^{2} + 19 x - 30 = 0 \] Now, to factor the quadratic \( 5 x^{2} + 19 x - 30 \), we look for two numbers that multiply to \( 5 \times -30 = -150 \) and add up to \( 19 \). The numbers 25 and -6 fit the bill since \( 25 \times -6 = -150 \) and \( 25 + (-6) = 19 \). We rewrite the middle term using these numbers: \[ 5 x^{2} + 25 x - 6 x - 30 = 0 \] Next, we group terms: \[ (5 x^{2} + 25 x) + (-6 x - 30) = 0 \] Factoring by grouping, we have: \[ 5x(x + 5) - 6(x + 5) = 0 \] Now, factoring out the common factor \((x + 5)\): \[ (5x - 6)(x + 5) = 0 \] Setting each factor to zero gives us the solutions: 1. \( 5x - 6 = 0 \) → \( x = \frac{6}{5} \) 2. \( x + 5 = 0 \) → \( x = -5 \) Thus, the solutions to the equation are: \[ x = \frac{6}{5} \quad \text{and} \quad x = -5 \] Let's not forget, math has a rich history! Quadratic equations have been studied since ancient times, with solutions dating back to Babylonian scribes around 2000 BC! They used geometric methods to solve these equations long before we had algebra as we know it today. As for applications, factoring quadratics isn't just academic—it's vital in fields like physics for projectile motion calculations, economics for maximizing profits, and even in computer graphics for rendering curves! Understanding these methods enables solution finding in various real-world scenarios.