1) Factor. \( 3 f^{2}+19 f+16 \)

To factor \(3f^2 + 19f + 16\), we look for two numbers that multiply to \(3 \times 16 = 48\) and add to \(19\). Those numbers are \(3\) and \(16\). Rewrite the middle term: \[ 3f^2 + 3f + 16f + 16 \] Now, group the terms: \[ (3f^2 + 3f) + (16f + 16) \] Factor out the common terms: \[ 3f(f + 1) + 16(f + 1) \] Now factor out \(f + 1\): \[ (3f + 16)(f + 1) \] So, the factored form of \(3f^2 + 19f + 16\) is \((3f + 16)(f + 1)\). Did you know that factoring quadratics like this has been a fundamental part of algebra for centuries? Ancient civilizations, including the Babylonians, were already manipulating polynomial equations, although their methods weren't as refined as our modern algebraic techniques. Factoring allows for easier problem-solving and simplification, making it an essential skill in mathematics! In real-world applications, factoring expressions is vital in fields such as engineering and physics. For instance, when analyzing the trajectory of projectiles or optimizing resource allocation in operations research, the ability to factor polynomials quickly allows professionals to devise more efficient solutions. So next time you factor, remember you're engaging in a skill that's foundational to solving complex real-world problems!