If 1 < x < y < z, which of the following has the greatest value?

Yeah really good question. If you distribute the multiplication you can get really direct comparisons between the quantities:

(A) zx + z

(B) zy + z

(C) xy + xz

(D) yx + yz

(E) zx + zy

(A) and (B) should be easy to eliminate since in each case you can see that choice (E) includes one of the two terms (zx or zy) but then adds something bigger than z to it.

Then for (C) vs. (D), note that each includes xy, but then (D) adds a bigger second term (since y > x, then yz > xz). So (D) is bigger than (C). And when you compare (E) to (D), you can use the same logic: each includes a yz term, but then (E) adds something bigger to it (zx > yx since z > y). So now you have algebraic proof that (E) is greater than all the other four.