Expanding and Factoring the Expression (x+3)(x3)(x+4)(x4)
This expression involves a pattern that allows for efficient expansion and factorization:
The Difference of Squares Pattern:
The difference of squares pattern states that: (a + b)(a  b) = a²  b²
Applying the pattern to the expression:
Notice that the expression contains two pairs of factors that follow the difference of squares pattern:
 (x + 3)(x  3)
 (x + 4)(x  4)
Expanding the expression:

First Pair: (x + 3)(x  3) = x²  3² = x²  9

Second Pair: (x + 4)(x  4) = x²  4² = x²  16

Combining the results: (x²  9)(x²  16)

Expanding further: x⁴  16x²  9x² + 144 = x⁴  25x² + 144
Factoring the expression back:

Recognize the pattern: The expanded form (x⁴  25x² + 144) resembles a quadratic equation if we substitute y = x²: y²  25y + 144

Factor the quadratic: (y  16)(y  9)

Substitute back: (x²  16)(x²  9)

Apply the difference of squares pattern again: (x + 4)(x  4)(x + 3)(x  3)
Therefore, we have successfully expanded and factored the given expression. The final factored form is (x + 4)(x  4)(x + 3)(x  3).
Key takeaways:
 The difference of squares pattern is a powerful tool for simplifying and factoring expressions.
 Recognizing patterns in expressions helps streamline the expansion and factorization process.
 By substituting variables, we can simplify expressions and make them easier to work with.