Write the following expression in factored form using the greatest common factor.
50a3 + 10a2

The factors of  50a³  are  1,  2,  5,  10,  25,  50,
   and their products with a, a² and a³ .

The factors of  10a²  are  1,  2,  5,  10,
    and their products with 'a' and a² .

Their common factors are  1, 2, 5, 10,
       and their products with 'a' and a².

Their greatest common factor is  10a² .

(Another way to spot it, easily, is to remember this helpful factoid:

                 If the smaller number is a factor of the larger number,
                 then the smaller number is their greatest common factor.

Using the greatest common factor, then . . .

           50a³ + 10a²  =     10a²(5a + 1) .

Iam unsure about what the proof is, therefore as much as wish to i can not.