prove that of sum square root of 2 and square root of 3 is not rational

prove that the square root of 2 plus the square root of 3 is not rational?

does always the sum of two not rational numbers is a not rational number?

i know the proof 2 = a^2/b^2

i separately proved that square root of 2 and square root of 3 are irrational

how two prove that the sum of two such numbers is irrational too?

i will try to prove by contradiction:

suppose

(2)^0.5 + (3)^0.5 is a rational number

(if we multiply a rational number by a rational number we will get a rational number "h")

5+2*(2)^0.5 * (3)^0.5=h

24=h^2 -10*h +25

h^2 -10*h +1=0

by this rational roots theorem

the possible roots is +1 and -1

not one of the represent the actual roots of h^2 -10*h +1=0

what is the next step in the prove?