What number goes in the question mark so that the quadratic is the square of a binomial? 9x^2 +6x+?

(3x+1)^2 =   9x^2 + 6x + 1

Let  m  be the mystery number.

9x² + 6x + m  =  (3x)² + 2(3x) + m

Now we can substitute  a  for  3x  like this:

a² + 2a + m

Now we don't have to worry about the coefficient of  9  on the first term....and so  m  is half the coefficient of the middle term squared.

m  =  ((1/2) * 2)^2   =   1^2   =   1

Check:

9x² + 6x + 1   =   (3x + 1)(3x + 1)   =   (3x + 1)²