I believe that the question stem should read as follows.
Note the change in red:

In the XY-coordinate plane, line L and line K intersect at the point (4, 3). Is the product of their slopes negative?

(1) The product of the x-intercepts of line L and K is positive.
(2) The product of the y-intercepts of line L and k is negative.

Don't solve. Draw.

Statement 1: The product of the x-intercepts of line L and K is positive.
K has a negative x-intercept, L has a negative x-intercept, the product of the slopes is positive:


K has a positive x-intercept, L has a positive x-intercept, the product of the slopes is negative:

INSUFFICIENT.

Statement 2: The product of the y-intercepts of line L and k is negative.
K has a negative y-intercept, L has a positive y-intercept, the product of the slopes is negative:


K has a negative y-intercept, L has a positive y-intercept, the product of the slopes is positive:

INSUFFICIENT.

Statements 1 and 2 combined:
Statement 1 requires that both x-intercepts be negative or that both x-intercepts be positive (so that their product is positive).
If both x-intercepts are negative, then both y-intercepts must be positive, which does not satisfy statement 2:


Thus, both x-intercepts must be positive.
Statement 2 requires that one of the y-intercepts be positive, the other negative.
A positive x-intercept and a positive y-intercept yields a negative slope.
A positive x-intercept and a negative y-intercept yields a positive slope.
See below:

Thus, the product of the slopes must be negative.
SUFFICIENT.

The correct answer is C.