I need to factor (my last post was incorrect)

24x^2 + 42x + 15

=3(8x^2 +14x +5)

=3(2x + 1)(4x +5)

24x^2+42x+15=3(8x^2+14+5)

=3(8x^2+10x+4x+5)
3[2x(4x+5)+1(4x+5)]
=3(4x+5)(2x+1)

Please read second expression as

3(8x^2+14x+5)

To factor the expression 24x^2 + 42x + 15, we need to look for two binomials that, when multiplied together, result in the given expression.

The first step is to find two numbers a and b such that their product is equal to the product of the coefficients of x^2 (24) and the constant term (15) and their sum is equal to the coefficient of x (42).

In this case, the product of 24 and 15 is 360. We need to find two numbers that multiply to 360 and add up to 42.

Next, we need to factorize 360. We can do this by finding the prime factorization of 360, which is 2^3 * 3^2 * 5.

Now we can try different combinations of those factors to find two numbers that add up to 42. Let's check:

1. (2^3 * 3) + (3^2 * 5) = 24 + 45 = 69 (not equal to 42)
2. (2^3 * 5) + (3^2) = 40 + 9 = 49 (not equal to 42)
3. (2^2 * 3 * 5) + (2 * 3^2) = 60 + 18 = 78 (not equal to 42)
4. (2^2 * 3^2) + (2 * 3 * 5) = 36 + 30 = 66 (not equal to 42)

None of these combinations give us the sum of 42. Therefore, it is not possible to factorize 24x^2 + 42x + 15 using whole numbers.

However, we can still factorize it using the quadratic formula or completing the square method. Applying the quadratic formula, we have:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For 24x^2 + 42x + 15, a = 24, b = 42, and c = 15. Substituting these values into the quadratic formula:

x = (-(42) ± sqrt((42)^2 - 4 * 24 * 15)) / (2 * 24)
x = (-42 ± sqrt(1764 - 1440)) / 48
x = (-42 ± sqrt(324)) / 48
x = (-42 ± 18) / 48

This gives us two possible solutions:
x = (-42 + 18) / 48 = -24/48 = -1/2
x = (-42 - 18) / 48 = -60/48 = -5/4

So, the factored form of 24x^2 + 42x + 15, considering the quadratic formula, is (2x + 1/2)(3x + 5/4).