Factor. \( 6 a^{2}-7 a-3 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( 6 a^{2}-7 a-3=\square \) \( \quad \) (Factor completely.) B. The polynomial is prime.

To factor the polynomial \( 6a^{2}-7a-3 \), we look for two numbers that multiply to \( 6 \times -3 = -18 \) and add to \( -7 \). The numbers \(-9\) and \(2\) fit the bill perfectly! We can then rewrite the middle term: \[ 6a^{2}-9a+2a-3 \] Now, group the terms: \[ (6a^{2}-9a)+(2a-3) \] Factoring out what we can gives us: \[ 3a(2a-3)+1(2a-3) \] Combining these yields: \[ (3a+1)(2a-3) \] So we conclude that: \[ 6a^{2}-7a-3=(3a+1)(2a-3) \] When dealing with polynomials, it's important to remember that patience is key. The process of factoring is like untangling a knot; it requires the right approach, and sometimes you might miss a detail. Common mistakes include overlooking negative signs or miscalculating the product and sum of coefficient pairs. Always double-check your work!