To factor the quadratic expression \(3x^2 - 8x + 5\) completely, we can use the method of finding two numbers that multiply to the product of the leading coefficient (3) and the constant term (5), which is \(3 \times 5 = 15\), and also add up to the middle coefficient (-8).
The two numbers that satisfy these conditions are -3 and -5 since:
- Multiplying: \(-3 \times -5 = 15\)
- Adding: \(-3 + -5 = -8\)
Now we can rewrite the middle term \(-8x\) using -3 and -5: \[ 3x^2 - 3x - 5x + 5 \]
Next, we group the terms: \[ (3x^2 - 3x) + (-5x + 5) \]
Now we factor out the common factors in each group: \[ 3x(x - 1) - 5(x - 1) \]
Since \((x - 1)\) is common in both groups, we can factor it out: \[ (3x - 5)(x - 1) \]
Thus, the complete factorization of the expression \(3x^2 - 8x + 5\) is: \[ \boxed{(3x - 5)(x - 1)} \]