31) If \( \triangle A B C \sim \triangle D E F \) and if \( A B=x^{2}-6 x-40 \) and \( D E=x^{2}-100 \), then what is the scale factor (in simplest form) of these similar triangles?

To find the scale factor between the triangles, we need to set up the ratios of the corresponding sides. Given that \( AB = x^2 - 6x - 40 \) and \( DE = x^2 - 100 \), we can express the scale factor \( k \) as: \[ k = \frac{AB}{DE} = \frac{x^2 - 6x - 40}{x^2 - 100} \] Next, we can factor both expressions. The quadratic \( AB \) can be factored as follows: \[ x^2 - 6x - 40 = (x - 10)(x + 4) \] While the quadratic \( DE \) factors into: \[ x^2 - 100 = (x - 10)(x + 10) \] Now, substituting these factored forms back into our scale factor expression gives: \[ k = \frac{(x - 10)(x + 4)}{(x - 10)(x + 10)} \] Assuming \( x \neq 10 \) (since we can't divide by zero), the \( (x - 10) \) terms cancel out: \[ k = \frac{x + 4}{x + 10} \] Thus, the scale factor \( k \) in simplest form is: \[ \frac{x + 4}{x + 10} \] This represents the ratio of the lengths of the two triangles based on the side lengths given.