Illegal Separation in Math

Recently, a high school junior in our Center took an assessment with us and made an error on the test that made half of the Instructors gasp and the other half shudder.  The student was a bit surprised that the problem was marked as incorrect but after it was explained to him in a non-traditional way, he understood.

Let's look at the error, shall we?

As you can see the binomial (x+2) is in the top of the fraction bar.  The polynomial (x^2-x+2) is in the bottom of the fraction bar. The directions were to simplify.  Just simplify.

Notice the steps the student took.  He crossed out the x from the numerator and the 2nd x term in the denominator.  He also crossed out the 2 in the numerator with the 2 in the denominator. This leaves the "+" operation suddenly alone when before it nicely joined the x and 2.  Crossing out the 2 from the denominator left the " - " sign hanging awkwardly at the end of the polynomial.  What is that "-" to do by itself?  There are o directions.  Subtract what?  The poor sign has no purpose now.

We tried explaining the error the traditional way but it didn't make sense to him so we tried a non-traditional way, as shown in the picture.  It's an ILLEGAL SEPARATION of a happy couple, "x and 2" And the operations were left with no purpose.  Factoring is your friend. Factor so that you can keep the couple together and make one by factoring out the same couple from the denominator.  

He asked if this was a political commentary.  Absolutely not!