To factor the quadratic expression 6x² + 11x + 10, we will first look for two numbers that multiply to the product of the coefficient of the quadratic term (6) and the constant term (10), which is 60 (6 * 10 = 60). At the same time, these two numbers must also add up to the coefficient of the linear term (11).
After analyzing the factors of 60, we find that the pair 5 and 6 meet our criteria since:
- 5 * 6 = 30
- 5 + 6 = 11
Next, we will rewrite the linear term using these two numbers:
6x² + 6x + 5x + 10
Now, we can group the terms:
(6x² + 6x) + (5x + 10)
Factoring out the common terms from each group gives us:
- From the first group, we can factor out 6x, leading to 6x(x + 1).
- From the second group, we can factor out 5, giving us 5(x + 2).
The expression can now be written as:
6x(x + 1) + 5(x + 2)
The final step is to factor out the common binomial (x + 2). Thus, the expression becomes:
3(2x + 5)(x + 2)
So, the factored form of the expression 6x² + 11x + 10 is:
3(2x + 5)(x + 2)