Factoring Trinomials that have Negatives

Algebra tiles are a great help when factoring trinomials.

We looked at a few.  The first question happened to be impossible, but now we know why.  We also were able to modify the question to create other options that are similar that do factor.

We are building rectangles out of these expressions.

Here’s the start of my rectangle.  I look at different ways I can arrange 14 that make rectangles (either 1×14 or 2×7).  I put my units in their rectangular arrangement diagonal to the x squared.

Next I try to fill in the gaps in the rectangle using my x rectangles.

At this point the rectangle is not full, so I’ll add some zero pairs of xs.  I know I need to put the positives so that they will multiply with the negatives to result in negatives in the bottom right corner. 

My rectangle still isn’t full, and if I had arranged my squares 1×14 it would make another rectangle that wouldnt be complete.  We know that this is an impossible question,  but that we could easily make a rectangle out of the following, which simplifies to x^2-2x-8

Or the following, which simplifies to x^2-5x-14.

After a lot of exploring we noticed something that is always true about the numbers in the rectangle and the numbers in the expression.  We are looking for a way to form the constant term into a rectangle with dimensions that will add up to the x coefficient.