Step 1 :Find two numbers that multiply to \(6 \times -5 = -30\) and add to -13. These numbers are -15 and 2.
Step 2 :Rewrite the original expression, splitting the -13x term into -15x and 2x: \(6x^2 - 15x + 2x - 5\).
Step 3 :Group the terms and factor out the greatest common factor from each group: \(3x(2x - 5) + 1(2x - 5)\).
Step 4 :Factor out the common factor \((2x - 5)\) to get the factorization: \((2x - 5)(3x + 1)\).
Step 5 :Check the work by expanding the product to see if we get the original expression: \((2x - 5)(3x + 1) = 6x^2 - 15x + 2x - 5 = 6x^2 - 13x - 5\).
Step 6 :So, the factorization of \(6x^2 - 13x - 5\) is \(\boxed{(2x - 5)(3x + 1)}\).