Final answer:
The polynomial 2x² + 7x + 6 is factorized into (x + 2)(2x + 3) by splitting the middle term, factoring by grouping, and then factoring out the common binomial (2x + 3).
Step-by-step explanation:
The factorization of the polynomial 2x² + 7x + 6 involves finding two binomials that multiply together to give the original quadratic polynomial. This process is called factoring by grouping or using the AC method, where A is the coefficient of the x² term and C is the constant term.
To factor 2x² + 7x + 6, we are looking for two numbers that multiply to A×C (2×6 = 12) and add up to B (7). These numbers are 3 and 4, because 3×4=12 and 3+4=7.
The next step is to split the x term (7x) into two terms using the numbers we found: 2x² + 3x + 4x + 6. Now, we can factor by grouping:
- First group: x (2x + 3)
- Second group: 2 (2x + 3)
Since 2x + 3 is common in both groups, we factor it out to get the final factorized form: (x + 2)(2x + 3).