Simplify the following algebraic expressions: 1) 4a² - b² + 12a + 6b 2) 64x² - 16xy + y² - 9

Solution:

• 1) 4a² - b² + 12a + 6b
  This expression can be factored by grouping. We can rearrange the terms to group similar variables together: (4a² + 12a) + (-b² + 6b).
  Factor out the common terms from each group: 4a(a + 3) - b(b - 6).
  At this point, it does not appear to factor further into a simple form without additional information or context suggesting a specific type of factorization (like a difference of squares or a perfect square trinomial applied to a portion of the expression). If this were intended to be a perfect square, it would likely have a structure like (2a ± b)². However, the terms +12a and +6b do not fit this pattern directly.
• 2) 64x² - 16xy + y² - 9
  This expression can be recognized as a combination of a perfect square trinomial and a difference of squares.
  First, group the terms that form a perfect square: (64x² - 16xy + y²). This is the expansion of (8x - y)².
  So, the expression becomes (8x - y)² - 9.
  This is now in the form of a difference of squares, a² - b², where a = (8x - y) and b = 3.
  The difference of squares factors as (a - b)(a + b).
  Substituting our values: ((8x - y) - 3)((8x - y) + 3).
  Therefore, the factored form is (8x - y - 3)(8x - y + 3).

Final Answer:

1) 4a² - b² + 12a + 6b (Cannot be easily factored into a simple form)

2) (8x - y - 3)(8x - y + 3)