The ellipse is $$3x^2 + 4y^2 = 12$$
$$\implies \displaystyle{\frac{x^2}{4} + \frac{y^2}{3} = 1}$$
$$\implies \displaystyle{\frac{x^2}{2^2} + \frac{y^2}{(\sqrt3)^2} = 1}$$
Here, semi-major axis $a = 2$ and semi-minor axis $b = \sqrt3$.
One of the foci can be calculated as $c^2 = a^2 - b^2 = 1 \implies c = 1$.
We know that the sum of the distances of the foci from any point on the ellipse is equal to $2a$.
So, $|PS| + |PS'| = 2a$. The line segments $\overline {PS}$, $\overline {PS'} $, $\overline {SS'} $ form a triangle.
So, $|PS| + |PS'| + |SS'| = 2a + 2c = 2*2 + 2*1 = 6$.
Answer is Option $(B)$.