\[ \begin{array}{l}3 x^{2}+30 x+63 \\ \text { Select the correct choice below and fill in any answer boxes within your choice. } \\ \text { Ax completely. } \\ \text { (Factor completely.) } \\ \text { B. The polynomial is prime. }\end{array} \]

To factor \( 3x^2 + 30x + 63 \) completely, first, we can factor out the greatest common factor, which is 3. This gives us: \[ 3(x^2 + 10x + 21) \] Next, we need to factor the quadratic \( x^2 + 10x + 21 \). We look for two numbers that multiply to 21 and add to 10. The numbers 3 and 7 fit the bill. Thus, we can rewrite the polynomial as: \[ 3(x + 3)(x + 7) \] The expression \( 3(x + 3)(x + 7) \) is the completely factored form of the original polynomial. From a historical insight, factoring polynomials dates back to ancient civilizations. The Babylonians and Greeks utilized geometric methods to solve quadratic equations, laying the groundwork for algebra as we know it today, including factoring techniques that are essential in higher mathematics. In terms of real-world application, understanding how to factor polynomials is crucial in various fields like physics, engineering, and economics. For example, it helps in solving equations that describe motion, optimizing functions for maximum profit in economics, or analyzing relationships between different variables in simulations and modeling.