Question

Express the following as the difference of two squares: 

(x² + 5x + 6) (x² – 5x – 6)

Answer 1

Given: (x² + 5x + 6) (x² – 5x – 6)

Step-wise calculation:

1. First, notice that both factors can be seen as conjugates but rearranged:

(x² + 5x + 6) and (x² – 5x – 6)

2. Observe that if you treat the second factor’s terms related to the first factor’s, you get a structure like:

(x²) + (5x + 6) and (x²) – (5x + 6)

3. Then the original product can be rewritten as:

(x² + 5x + 6)(x² – 5x – 6)

= (x² + (5x + 6))(x² – (5x + 6))

4. Recognize this as the difference of two squares formula:

(a + b)(a – b) = a² – b²

where a = x² and b = 5x + 6.

5. Therefore, (x² + 5x + 6)(x² – 5x – 6) = (x²)² – (5x + 6)².