Factor the trinomial. \( 20 \mathrm{a}^{2}-23 \mathrm{ad}+6 \mathrm{~d}^{2} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( 20 \mathrm{a}^{2}-23 a d+6 \mathrm{~d}^{2}=\square \) (Factor completely.) B. The trinomial is prime.

To factor the trinomial \( 20 \mathrm{a}^{2}-23 \mathrm{ad}+6 \mathrm{~d}^{2} \), we can look for two numbers that multiply to \( 20 \cdot 6 = 120 \) and add up to \(-23\). The numbers \(-15\) and \(-8\) fit this requirement. We can regroup the middle term and factor by grouping: \[ 20 \mathrm{a}^2 - 15 \mathrm{ad} - 8 \mathrm{ad} + 6 \mathrm{d}^2. \] This can be grouped as: \[ 5\mathrm{a}(4\mathrm{a}-3\mathrm{d}) - 2(4\mathrm{a}-3\mathrm{d}). \] Factoring out the common term, we get: \[ (5\mathrm{a}-2)(4\mathrm{a}-3\mathrm{d}). \] So, the complete factorization is: A. \( 20 \mathrm{a}^{2}-23 a d+6 \mathrm{~d}^{2}=(5\mathrm{a}-2)(4\mathrm{a}-3\mathrm{d}) \). --- Did you know that factoring polynomials dates back to ancient civilizations? Babylonian mathematicians already had sophisticated ways of solving quadratic equations, laying the groundwork for modern algebra! Their methods were intuitive yet effective, showcasing the early fascination with patterns and relationships in numbers. When it comes to applications, factoring trinomials is vital in various fields, including physics and engineering, where you frequently encounter quadratic equations. Whether it’s calculating projectile motions or optimizing designs, being able to factor efficiently can simplify complex problems into manageable tasks. Plus, it’s a handy skill for analyzing functions in real-world situations!