Final answer:
Factorizing the quadratic equation 2x² + 5x - 3 involves finding two numbers that multiply to -6 (the product of the coefficients of 2x² and -3) and add up to 5 (the coefficient of x). The numbers that satisfy this are 6 and -1. Factoring by grouping, the equation can be factorized as (2x - 1)(x + 3).
Step-by-step explanation:
To factorize the quadratic equation 2x² + 5x - 3, we are looking for two binomials that will multiply to give us this original expression. A method we can use is to seek two numbers that multiply to give us the product of the coefficients of 2x² (which is 2) and the constant term (which is -3), so this product is -6, but at the same time, these two numbers must add up to the coefficient of the middle term x, which is 5.
These two numbers are 6 and -1 because 6 multiplied by -1 gives -6, and 6 added to -1 gives 5, which is the coefficient of the middle term. Thus, the original expression can be written as 2x² + 6x - x - 3. Now we can factor by grouping: 2x can be factored out of the first two terms, and -1 can be factored out of the last two terms, resulting in 2x(x + 3) - 1(x + 3). Since (x + 3) is common in both terms, we can further factorize to get the final answer, (2x - 1)(x + 3), which corresponds to option (a).