Final answer:
To factorize 12g² - 12g - 45, use the method of grouping. The factorization is (2g - 5)(6g + 9).
Step-by-step explanation:
To factorize the quadratic trinomial 12g² - 12g - 45, we need to find two binomials that when multiplied together, give us the original trinomial. One way to do this is by using the method of grouping.
Step 1:
Multiply the coefficient of the leading term (12) and the constant term (-45) to get -540. We are looking for two factors that multiply to -540 but add up to the coefficient of the middle term (-12).
The factors of -540 that add up to -12 are -30 and 18.
Step 2:
Split the middle term (-12g) into two terms using the factors found in Step 1:
12g² - 30g + 18g - 45
Step 3:
Factor by grouping. Group the first two terms and the last two terms:
(12g² - 30g) + (18g - 45)
Step 4:
Factor out the greatest common factor (GCF) from each group:
6g(2g - 5) + 9(2g - 5)
Step 5:
Notice that we have a common binomial factor (2g - 5) in both groups. Factor it out:
(2g - 5)(6g + 9)
Therefore, the factorization of 12g² - 12g - 45 is (2g - 5)(6g + 9).
Learn more about Factoring quadratic trinomials