$$\sum_k^n P_k x_k$$
Am I allowed to split it up into two sums so I have it like
$$\sum_k^n P_k \sum_k^nx_k$$
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$$x_1\qquad\qquad\qquad\qquad x_2\qquad\qquad\qquad\qquad x_3\qquad\qquad\qquad\qquad x_4$$
Imagine that the vertical segments are your $P$'s. Then the product of sums is the area of the entire rectangle, whereas the sum of products is only the red area.
Assuming P depends on k and not n then the summation $$ \sum_k P_k x_k $$ Creates terms like $$ P_1 x_1 + P_2 x_2 + \ldots + P_n x_n $$ But, if you took $P_k$ out then you would have two independent sums: $$ \sum_k P_k \sum_k x_k $$ Producing two separate sums multiplying each other like $$ (P_1 + P_2 + \ldots + P_n) (x_1 + x_2 + \ldots + x_n) $$ And this obviously produces terms like $P_1 x_1, P_1 x_2,\ldots$ or as produced by the double sum written like: $$ \sum_i \sum_j P_i x_j $$ So the original summation is definitely not the same as this double sum.
^^^That offers a nice summary of summation notation and its properties.
To answer your question specifically, you should not split this into two different summations. Rather, notice that P does not depend on k. You can treat P as a constant and simply move it outside the summation so that it multiples the result of your summation of $x_k$, which does depend on your variable k.