I'm factoring special cases and i cant figure this problem out:

45t^2-72t+24

please help

45t^2 - 72t + 24

Only 3 can be factored from each term.

3(15t^2-24t+8)

To factor the expression 45t^2 - 72t + 24, we can start by looking for common factors among the terms. In this case, we can divide all the terms by their common factor of 3 to simplify the expression.

Dividing each term by 3, we get:
45t^2 / 3 - 72t / 3 + 24 / 3
which simplifies to:
15t^2 - 24t + 8

Now let's consider how we can factor the simplified expression further.

Step 1: Look for common factors among the terms (if any). In this case, all the terms have a common factor of 1, so we can skip this step.

Step 2: Look for two numbers that multiply to give the product of the coefficient of the quadratic term (15) and the constant term (8), and add up to give the coefficient of the linear term (-24).

In this case, the numbers are -3 and -5 because:
-3 * -5 = 15 (product of coefficient of quadratic term and constant term)
-3 + (-5) = -8 (coefficient of linear term)

Step 3: Rewrite the middle term (-24t) using the two numbers found in Step 2.

15t^2 - 3t - 5t + 8

Step 4: Group the terms with a common factor.

(15t^2 - 3t) + (-5t + 8)

Step 5: Take out the common factors from each group.

3t(5t - 1) - 8(5t - 1)

Step 6: Notice that we now have a common factor of (5t - 1).

(3t - 8)(5t - 1)

So, the factored form of the expression 45t^2 - 72t + 24 is (3t - 8)(5t - 1).

To factor the quadratic expression 45t^2 - 72t + 24, we first check if there is a common factor among all the terms. In this case, we can divide all the terms by 3:

45t^2 / 3 - 72t / 3 + 24 / 3

This simplifies to:

15t^2 - 24t + 8

Next, we can try to factor this quadratic expression further. The first term, 15t^2, can be factored as 3t * 5t, and the last term, 8, can be factored as 2 * 4. Now, let's try to find two numbers whose product is 2 * 4 = 8 and whose sum is the coefficient of the middle term, -24.

We can see that -12 and -2 satisfy both conditions, since -12 * -2 = 24 and -12 + (-2) = -24. Using these numbers, we rewrite the middle term -24t as -12t - 12t:

15t^2 - 12t - 12t + 8

Next, we group the terms:

(15t^2 - 12t) + (-12t + 8)

Now, we factor out the greatest common factor from each group:

3t(5t - 4) - 4(3t - 2)

Finally, we have factored the quadratic expression:

3t(5t - 4) - 4(3t - 2)