To find the complete factorization of the given expression 10x^3 - 35x^2 - 20x, we first look for the common factors.
Looking at the coefficients 10, -35, and -20, we can see that 5 can be a common factor. Also, each term has at least one 'x', so x can be a common factor as well.
Therefore, we can factor out 5x from each term to get:
5x(2x^2 - 7x - 4).
Next, we need to factor the resulting quadratic 2x^2 - 7x - 4. We are looking for two numbers that multiply to (2*-4)=-8 and add up to -7. These two numbers are -1 and 8.
After rewriting the middle term we get:
5x(2x^2 - 8x + x - 4).
Now, we can factor by grouping:
5x[(2x^2 - 8x) + (x - 4)].
Inside the parentheses, factor out 2x from the first two terms and 1 from the last two:
5x[2x(x - 4) + 1(x - 4)].
Notice that (x-4) is a common term, factor it out:
5x[(2x + 1)(x - 4)].
Now, we can see that the expression 10x^3 - 35x^2 - 20x factors to 5x(2x + 1)(x - 4).