Factor the trinomial, \( 12 q^{2}-29 q d+14 d^{2} \) Solect the correct choice below and, if necessary fill in the answer box lo cormplete your choice. A. \( 12 q^{2}-29 q d+14 d^{2}=\square \) (Factor completely.) B. The trinomial is prime

To factor the trinomial \( 12 q^{2}-29 q d+14 d^{2} \), we can look for two numbers that multiply to \( 12 \times 14 = 168 \) and add up to \(-29\). Those numbers are \(-21\) and \(-8\). We can then rewrite the middle term and factor by grouping: \[ 12 q^{2} - 21 q d - 8 q d + 14 d^{2} = 3q(4q - 7d) - 2d(4q - 7d) = (3q - 2d)(4q - 7d) \] So, the complete factored form is \( 12 q^{2}-29 q d+14 d^{2} = (3q - 2d)(4q - 7d) \). Now, let's talk about the importance of knowing how to factor such expressions! When you factor trinomials, you're not just playing with numbers; you’re also preparing yourself for solving quadratic equations and even diving into calculus. It'll save you time and effort when finding roots or simplifying expressions! And while we're on the topic, remember to watch out for sign mistakes and common factoring errors. Double-check your terms and make sure that when you multiply back, you’re actually getting your original trinomial! It’s super easy to misplace a negative sign or omit a term accidentally. Happy factoring!