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To solve the given expression, we need to add the two fractions:

\[
\frac{2}{3x^2 - 10x + 3} + \frac{1}{x - 3}
\]

### Steps to simplify:

1. **Factor the denominator** of the first fraction: \( 3x^2 - 10x + 3 \).
   - Factor this quadratic expression using techniques like factoring by grouping or the quadratic formula.

2. **Find a common denominator** between the two fractions. The common denominator will be the least common multiple (LCM) of the factored form of \( 3x^2 - 10x + 3 \) and \( x - 3 \).

3. **Combine the fractions** over the common denominator.

4. **Simplify** the resulting expression if possible.

### Step 1: Factor the quadratic
We factor \( 3x^2 - 10x + 3 \):
- We look for two numbers that multiply to \( 3 \times 3 = 9 \) and add to \( -10 \). These numbers are \( -9 \) and \( -1 \).
- So we can factor as: 
  \[
  3x^2 - 10x + 3 = (3x - 1)(x - 3)
  \]

### Step 2: Rewrite the expression
Now the expression becomes:
\[
\frac{2}{(3x - 1)(x - 3)} + \frac{1}{x - 3}
\]

### Step 3: Find a common denominator
The common denominator is \( (3x - 1)(x - 3) \). Rewrite the second fraction to have this denominator:
\[
\frac{1}{x - 3} = \frac{3x - 1}{(3x - 1)(x - 3)}
\]

### Step 4: Add the fractions
Now that the denominators are the same, we can add the numerators:
\[
\frac{2}{(3x - 1)(x - 3)} + \frac{3x - 1}{(3x - 1)(x - 3)} = \frac{2 + (3x - 1)}{(3x - 1)(x - 3)}
\]

Simplify the numerator:
\[
2 + (3x - 1) = 3x + 1
\]

So the result is:
\[
\frac{3x + 1}{(3x - 1)(x - 3)}
\]

### Final Answer:
\[
\frac{3x + 1}{(3x - 1)(x - 3)}
\]

This is the simplified form of the sum of the two fractions.

Would you like any further details or clarifications on any of the steps?

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Here are five related questions for further practice:

1. How do you factor quadratics like \( 3x^2 - 10x + 3 \) in general?
2. What is the process for finding a common denominator in fraction addition?
3. How can you simplify complex rational expressions with quadratic terms?
4. What other methods can be used if factoring by grouping doesn't work?
5. What strategies can help verify if the simplification is correct?

**Tip**: Always check if a quadratic can be factored first, as it simplifies further operations like finding common denominators!

Add: (2 / (3x^2 - 10x + 3)) + (1 / (x - 3)). Simplify your answer as much as possible.

Learn how to add two rational expressions: (2 / (3x^2 - 10x + 3)) + (1 / (x - 3)), including factoring quadratic expressions and finding a common denominator. Suitable for high school algebra students in Grades 10-12.

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