To factorize the quadratic expression \(9x^2 - 21x + 10\), we follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 9\)
- \(b = -21\)
- \(c = 10\)

### Step 2: Find the product of \(a\) and \(c\)
Calculate the product of the first and last coefficients:
\[
ac = 9 \times 10 = 90
\]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(90\) and add to \(-21\).

The two numbers that satisfy these conditions are \(-15\) and \(-6\) because:
\[
-15 \times -6 = 90 \quad \text{and} \quad -15 + (-6) = -21
\]

### Step 4: Rewrite the middle term using the two numbers
Rewrite the expression by splitting the middle term \(-21x\) using \(-15x\) and \(-6x\):
\[
9x^2 - 15x - 6x + 10
\]

### Step 5: Factor by grouping
Group the terms in pairs:
\[
(9x^2 - 15x) - (6x - 10)
\]

Factor out the common factors from each group:
\[
3x(3x - 5) - 2(3x - 5)
\]

### Step 6: Factor out the common binomial
Notice that \((3x - 5)\) is a common factor:
\[
(3x - 5)(3x - 2)
\]

So, the fully factorized form of \(9x^2 - 21x + 10\) is:
\[
\boxed{(3x - 5)(3x - 2)}
\]

Would you like more details on any step, or do you have any questions?

### Related Questions
1. How can you use the quadratic formula to solve \(9x^2 - 21x + 10 = 0\)?
2. How do you determine if a quadratic expression can be factorized easily?
3. Can you explain how to factorize other types of quadratic expressions?
4. What are the steps to verify if the factorization is correct?
5. How does the discriminant help in understanding the factorization of quadratic equations?

### Tip:
Always check the factorization by expanding the factors to see if you get the original expression.

Discover how to fully factorize 9x^2 - 21x + 10 with a detailed step-by-step solution. This problem covers key algebraic concepts such as quadratic equations and factorization, using the factor theorem and quadratic formula. Suitable for students in Grades 7-9.

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