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Call the numbers x, y and suppose x>y
Their difference is 16 :x-y=16\implies\,y=x-16

Their sum is x+y and product is xy
Adding sum and product gives x+y+xy

Substitute the y value and get a quadratic in xx+y+xy=x+x-16+x(x-16)\\=x^2-14x-16
Minimum value occurs at vertex and the x coordinate of vertex is x=-\frac{b}{2a}=-\frac{-14}{2}=7

Plugging x in earlier equation gives y=7-16=-9
Therefore the required numbers are 7 and -9
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