what is the smallest of 3 consecutive positive integers if the product of the smaller two integers is 5 less than 5 times the largest intege

I think geno made a slight error here.......taking it from this point, we have

x^2 + x = 5x + 10 - 5

x^2 + x = 5x + 5     subtract 5x, 5 from both sides

x^2 - 4x - 5  = 0     factor

(x - 5) (x + 1) = 0    and setting each factor to 0 we have that x = 5  or x = -1

Reject  x= -1

So x = 5 ......x + 1 = 6 .... and x + 2 = 7

And......as a check...... (x)(x + 1)  + 5 = 5(x + 3)  →  (5)(6) + 5  = 5(7)  

And 35 = 35

Let the smallest integer be x.

Then, the next two consecutive integers are  x + 1  and  x + 2.

Multiplying the smaller two integers, you get  x(x + 1)  so the equation becomes:

   x(x + 1)  = 5(x + 2) - 5

    x² + x  =  5x + 10 - 5

    x² + x  =  5x += 5

    x² + x - 5x + 5  =  0

   x² - 4x + 5  =  0  

This can't be factored using whole numbers, so there is no positive integer answer.

If you use the quadratic formula, you get:  x  =  [4 ± √(-4)]/2,

which shows that there are no real answers.