Final answer:
The quadratic expression 18x²+27x-5 can be factored by the grouping method into (6x - 1)(3x + 5), which is confirmed by using FOIL to return to the original expression.
Step-by-step explanation:
To factor the quadratic expression 18x²+27x-5 by the grouping method, we first look for two numbers that multiply to give us (18 * -5 = -90) and add to give us 27. These two numbers are 30 and -3. We can then write the middle term as the sum of 30x and -3x.
The expression becomes 18x² + 30x - 3x - 5. Now, we group the terms: (18x² + 30x) + (-3x - 5). From the first group, we can factor out 6x, resulting in 6x(3x + 5). From the second group, we can factor out -1, resulting in -1(3x + 5).
Both groups now have a common factor of (3x + 5), so we can combine them: (6x - 1)(3x + 5).
To check the answer, we use the FOIL method (First, Outer, Inner, Last) to multiply the factors back out:
- First: 6x * 3x = 18x²
- Outer: 6x * 5 = 30x
- Inner: -1 * 3x = -3x
- Last: -1 * 5 = -5
Adding these results together, we get the original expression: 18x² + 27x - 5, confirming that our factored form is correct.