Question 1136473
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Sam    = 2*((Jon)^2 ) + 4*Jon.    (1)    ("Sam was twice the square of Jon's age plus another four times Jon's age")

Helen  =     Jon^2 + 45           (2)    (Second condition).


Since Sam and Helen are twins, their ages are the same; hence, right sides of equations (1) and (2) are equal:


    2*J^2 + 4J = J^2 + 45,    or


    J^2 + 4J - 45 = 0,

    (J-5)*(J+9) = 0.


Only positive root is meaningful:  J = 5.


Hence, Sam = 2*J^2 + 4*J = 2*5^2 + 4*5 = 2*25 + 20 = 50 + 20 = 70.


<U>ANSWER</U>.  Sam is 70 years old.
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You can check my answer on your own by calculating the Helen's age.