To factor the expression \( 20r^3 - 39r^2 + 18r \), we can follow these steps:

### Step 1: Factor out the greatest common factor (GCF).
The terms in the expression share a common factor of \( r \). So, we can factor out \( r \) from the entire expression:

\[
20r^3 - 39r^2 + 18r = r(20r^2 - 39r + 18)
\]

### Step 2: Factor the quadratic \( 20r^2 - 39r + 18 \).
Now, we need to factor the quadratic expression \( 20r^2 - 39r + 18 \). We look for two numbers that multiply to \( 20 \times 18 = 360 \) and add up to \( -39 \).

These numbers are \( -24 \) and \( -15 \), because:
\[
-24 \times -15 = 360 \quad \text{and} \quad -24 + -15 = -39
\]

### Step 3: Rewrite the middle term using these two numbers.
We rewrite the middle term \( -39r \) as \( -24r - 15r \):

\[
20r^2 - 39r + 18 = 20r^2 - 24r - 15r + 18
\]

### Step 4: Factor by grouping.
Group the terms in pairs:

\[
(20r^2 - 24r) - (15r - 18)
\]

Factor out the GCF from each group:

\[
4r(5r - 6) - 3(5r - 6)
\]

### Step 5: Factor out the common binomial.
Now, factor out the common factor \( (5r - 6) \):

\[
(4r - 3)(5r - 6)
\]

### Step 6: Combine the factors.
The fully factored form of the expression is:

\[
r(4r - 3)(5r - 6)
\]

Thus, the factorization of \( 20r^3 - 39r^2 + 18r \) is:
\[
r(4r - 3)(5r - 6)
\]

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Would you like more details or have any questions about this process?

Here are 5 related questions for further practice:
1. How would you factor \( 30r^3 - 45r^2 + 15r \)?
2. Can you factor \( 10r^3 - 29r^2 + 12r \) by the same method?
3. What is the greatest common factor (GCF) of \( 25r^3 - 50r^2 + 20r \)?
4. How can you check your factored form is correct by expanding?
5. How would the factorization change if the signs in the quadratic were different?

**Tip:** When factoring trinomials, always check for a greatest common factor first, as it simplifies the expression greatly!

Learn how to factor the expression 20r^3 - 39r^2 + 18r using the GCF and factoring by grouping. This problem involves key algebraic concepts such as factoring quadratics and the greatest common factor (GCF). Suitable for high school students in Grades 9-12.

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