Factor the following trinomaial completely. \( 6 x^{2}+7 x+2 \) Submit Question

To factor the trinomial \( 6x^{2}+7x+2 \), we look for two numbers that multiply to \( 6 \times 2 = 12 \) and add up to \( 7 \). The numbers \( 3 \) and \( 4 \) fit the bill! We can then rewrite the trinomial: \[ 6x^{2}+3x+4x+2 \] Next, we group the terms: \[ (6x^{2}+3x) + (4x+2) \] Factoring each group gives us: \[ 3x(2x+1) + 2(2x+1) \] Now, we can factor out the common term \((2x+1)\): \[ (2x+1)(3x+2) \] So, the completely factored form of the trinomial is: \[ (2x+1)(3x+2) \] --- Here's a fun tidbit: In ancient Egypt, mathematics was all about practical applications, like building pyramids. They didn't have algebra as we know it, but they were masters at balancing equations—much like factoring a trinomial! In modern times, factoring polynomials is crucial in fields like engineering and computer science. Whether designing algorithms or optimizing solutions, the concepts of simplifying expressions and factoring come directly into play, making them not just a classroom exercise but a fundamental skill in real-world applications!