Is it more convenient to solve the problem of solid geometry with space analytic geometry instead of vector method?

It can be replaced, but it is not necessarily more convenient. Just like solving equations with determinant, it is not necessarily better

Given two points a (- 2,0) and B (2,0), if the moving point m makes the product of the slopes of the lines Ma and MB equal to the constant - 3, then the trajectory equation of point m?

Let the coordinates of point m be (x, y)
Then (Y / (x + 2)) * (Y / (X-2)) = - 3
x^2/4+y^2/12=1

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