Why is the axiom for vector space not satisfied by the following equation?

Question

Vector sum $(x_1, x_2)+'(y_1, y_2)=(x_1+2y_1, 3x_2-2y_2)$ and the usual scalar multiplication $c(x_1, x_2)=(cx_1, cx_2)$.

Sure additive properties does not hold for the operation but why does the property $(c+d)x=cx+dx$ not hold either?

What did I miss?

Answer 1

Well you could try calculating the RHS of that expression, $$cx+dx=c(x_1,x_2)+d(x_1,x_2)=(cx_1,cx_2)+(dx_1,dx_2)=(cx_1+2dx_1,3cx_2-2dx_2),$$ compare it with the LHS which is $$(c+d)x=(c+d)(x_1,x_2)=\big((c+d)x_1,(c+d)x_2\big).$$ Are they the same?