lets name the smaller integer as x

the consecutive integer is x + 1 as its the next number

the product of these 2 numbers are 812

you get the product once you multiply the 2 numbers

x * (x + 1) = 812

x² + 1x = 812

x² + x - 812 = 0

this a quadratic equation

we have to first find the factors of 812 * x² with a difference or sum of + x

the 2 factors are +29x and - 28x which have a product of 812x² and difference of +x

we substitute +x in the quadratic equation with +29x - 28x

x² + 29x -28x - 812 = 0

x(x + 29) -28(x + 29) = 0

(x+ 29)(x-28) = 0

so there are 2 possibilites for x

x + 29 = 0 or x - 28 = 0

x = -29 x = 28

since we are asked to find positive integers from the 2 possible values of x

x = 28 is the positive number

therefore smaller integer is 28