Final answer:
To factor the trinomial 5x²+34x+24, find two numbers that add to 34 and multiply to 120. These are 10 and 12. Factoring by grouping, we get the final factored form as (5x + 12)(x + 2).
Step-by-step explanation:
The process of factoring the trinomial 5x²+34x+24 involves finding two numbers that both add to the middle coefficient, 34, and multiply to the product of the first and last coefficients, which is 5 times 24, equaling 120. When we break down the number 120, we are looking for a pair of factors that can be used to rewrite the middle term of 34x.
Upon examining the factors of 120, we find that 10 and 12 are the numbers that add up to 34. Therefore, we can split the middle term, 34x, into two terms using 10 and 12. The trinomial can now be expressed as 5x² + 10x + 24x + 24.
Next, we can factor by grouping. We group the terms as (5x² + 10x) and (24x + 24) and factor out the greatest common factor from each group. From the first group, we factor out 5x, and from the second group, we factor out 12. This gives us 5x(x + 2) + 12(x + 2). Notice that (x + 2) is a common factor.
Finally, we factor out the common factor (x + 2), which leaves us with the completely factored form of the trinomial: (5x + 12)(x + 2).