To factor the quadratic expression 15x² + 14x + 8, we need to look for two numbers that multiply to the product of the coefficient of x² (which is 15) and the constant term (which is 8), giving us 15 * 8 = 120, and that also add up to the coefficient of x (which is 14).
We need to find two numbers that meet the following criteria:
- Multiply to 120
- Add up to 14
After testing different pairs, we find that the numbers 10 and 12 work because:
- 10 * 12 = 120
- 10 + 12 = 14
Now, we can rewrite the middle term (14x) using these two numbers:
15x² + 10x + 4x + 8
Next, we can group the terms:
(15x² + 10x) + (4x + 8)
Factoring out the common factors from each group:
5x(3x + 2) + 4(3x + 2)
Now, we notice that both groups contain the common binomial factor (3x + 2), so we can factor that out:
(5x + 4)(3x + 2)
Finally, the factored form of 15x² + 14x + 8 is:
(5x + 4)(3x + 2)