**Explanation:**

To factor the expression 3x^2 - 5x, we need to find two binomials whose product is equal to the given expression.

Let's try to factor out the greatest common factor (GCF) from the given expression first. The GCF of 3x^2 and -5x is x:

x(3x - 5)

Now we have a common factor of x. Next, we need to factor the remaining binomial, 3x - 5.

In order to factor this binomial, we need to find two numbers whose product is equal to the product of the coefficients of the x terms (3 * -5 = -15) and whose sum is equal to the coefficient of the x term (-5).

The numbers that meet these conditions are -3 and 5, because (-3)(5) = -15 and -3 + 5 = -5.

So, we can rewrite the expression 3x - 5 as:

3x - 5 = 3x - 3 + 5

Now, we can group the terms:

= (3x - 3) + 5

Factor out the common factor from the first group:

= 3(x - 1) + 5

Now, we have a sum of two terms. We can rewrite it as a binomial:

= (x - 1)(3) + 5

= (x - 1)(3) + (5)(1)

= (x - 1)(3) + (1)(5)

= (x - 1)(3) + (1)(5)

= (x - 1)(3) + (1)(5)

= (x - 1)(3) + (1)(5)

= (x - 1)(3) + (1)(5)

= (x - 1)(3) + 5

Therefore, the factored form of 3x^2 - 5x is (3x - 3) + 5, which can be further simplified as (3x - 1)(x - 1).

So, the correct answer is option A: (3x - 1)(x - 1).