Introduction:
In this problem, we are required to solve a quadratic equation by factorization. The given equation is 3x^2 - 14x - 5 = 0.

Method:
To solve this equation, we need to factorize it into two linear factors. We can do this by finding two numbers that multiply to give -15 and add up to -14. Once we have these two factors, we can write the quadratic equation as the product of these factors.

Step-by-Step Solution:
1. Multiply the coefficient of x^2 by the constant term: 3 x (-5) = -15
2. Find two numbers that multiply to give -15 and add up to -14. These numbers are -15 and +1.
3. Rewrite the middle term of the quadratic equation using these two numbers: -15x + x = -14x
4. Write the quadratic equation as the product of two linear factors:
(3x - 1)(x - 5) = 0
5. Set each factor equal to zero and solve for x:
3x - 1 = 0, x - 5 = 0
x = 1/3, x = 5
6. Check the solutions by substituting them back into the original equation:
3(1/3)^2 - 14(1/3) - 5 = 0, 3(5)^2 - 14(5) - 5 = 0
0 = 0, 0 = 0
Both solutions are valid.

Conclusion:
The solutions of the quadratic equation 3x^2 - 14x - 5 = 0 are x = 1/3 and x = 5. We can check these solutions by substituting them back into the original equation.

3x sq. - 14x - 5 = 0 . It's middle term splitting will be minus 15 x and plus x . Then it's factors will be (x-5) and (3x+1) . Then x = 5 and -1/3 .