To solve, let's assign a variable that represents the age of the triplets this year. Let it be x.

Since the triplets are born 2 years after their sister Julie, then Julie is 2 years older than the triplets. So the expression that represents her age is x + 2.

So the age of the four siblings are:

Andrew = x, Alice = x, Alex = x, and Julie = x + 2.

To set-up the equation, express in math form the given condition that "the product of their four ages is 24,142 greater than the sum of their ages".

The math form for the product of their four ages is x*x*x*(x+2). And the math form for 24,142 greater than the sum of their ages is x+x+x+(x+2)+24,142. Setting these two expressions equal to each other, the resulting equation is

`x*x*x*(x+2) = x+x+x+(x+2)+24,142`.

Each side simplifies to the following:

`x^3(x+2)= 4x+ 24,144`

and

`x^4+2x^3=4x+24,144`.

Then, set one side equal to zero:

`x^4+2x^3-4x-24144=0`.

To solve for the values of x, graph the polynomial y=x^4+2x^3-4x-24144 using a graphing utility. From that graph, determine the values of x when y=0 (curve intersect with the x-axis).

(See also the attached graph.)

Base on that graph, the values of x when y=0 are

`x=12 and x =-12.99`.

Since x represents age, consider only the positive value. So, x=12. This is the age of the triplets Andrew, Alice, and Alex.

To get the age of Julie, plug in the value of x to the expression that represents the age of Julie.

Julie = x + 2 = 12 + 2 = 14.

Therefore, Julie is 14 years old.

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