The balanced equation tells us that, under ideal conditions (theoretical) we'll get 4 moles of PCl3 for every 1 mole of P4. The molar ratio is 4 moles PCl3/mole P4. Or, 0.25 moles P4/mole PCl3.
We want to wind up with 512g of PCl3, but we're told this reaction only has a 78% yield. Phooey. (the technical term for this level of yield). So let's plan ahead and adjust the amount of ending material we should target for by dividing the 512 grams by 0.78, which tells us we should target for 656.4 grams of PCl3, so that we wind up with (0.78*656.4) 512 grams.
656.4 grams of PCl3 is 4.78 moles of the material (656.4
g/137.3 g/mole). Multiply this by the molar ratio we deduced earlier (0.25 moles P4/PCl3) to find the moles P4 we need.
(0.25 moles P4/PCl3)*(4.78 moles PCl3) = 1.195 moles P4. Convert to grams by multiplying by the molar mass of P4. That gives us 148.1 grams P4. Slightly different than option 2, but that may be due to a difference in atomic masses or rounding.