Factor. \[ x^{2}-3 x y-4 y^{2} \]

Step 1 ：The given expression is a quadratic in terms of x. We can factorize it by finding two numbers that add up to -3y (the coefficient of x) and multiply to -4y^2 (the constant term).

Step 2 ：Let's denote these two numbers as a and b. So, we need to solve the following system of equations: a + b = -3y and a * b = -4y^2.

Step 3 ：We can solve this system of equations by either substitution or elimination method. However, since this is a quadratic equation, we can also use the quadratic formula to find the roots. The quadratic formula is given by: x = [-b ± sqrt(b^2 - 4ac)] / (2a).

Step 4 ：In this case, a = 1 (the coefficient of x^2), b = -3y, and c = -4y^2. Let's substitute these values into the quadratic formula to find the roots.

Step 5 ：These roots will be the values of x for which the given expression becomes zero. Hence, the factorized form of the given expression will be (x - root1)(x - root2).

Step 6 ：Let's calculate the roots and hence the factorized form of the given expression.

Step 7 ：The roots of the quadratic equation are 4y and -y. Hence, the factorized form of the given expression is (x - 4y)(x + y). This is the final answer.

Step 8 ：Final Answer: \(\boxed{(x - 4y)(x + y)}\)