Let's begin by using the variables 'x' and 'y' to represent the two numbers.

Next, let's write two algebraic equations to represent the problem.

For twice the sum of two numbers is 24, we can write:

2(x+y)=24 divide both sides by 2

x + y = 12

For the sum of the squares of the two numbers is 306, we can write:

x^(2)+y^(2)=306

The problem wants us to solve for the product of the two numbers, or xy.

Now notice that the key to solving this problem involves squaring equations.

Square the equation x + y = 12

(x+y)^(2)=12^(2)

x^(2)+2xy+y^(2)=144

Substitute the equation x^(2)+y^(2)=306 into the above equation:

306 + 2xy = 144

2xy = 144 - 306

2xy = -162 divide both sides by 2

xy = - 81

The product of the two numbers is -81.