Let's analyze and simplify each expression:

(a) $(a^2 + 4a + 4) - 9b^2$

1. Notice that $a^2 + 4a + 4$ can be factored as $(a + 2)^2$.
2. So, the expression becomes $(a + 2)^2 - (3b)^2$.
3. This is a difference of squares: $(a + 2 + 3b)(a + 2 - 3b)$.

Simplified Form:

$(a + 2 + 3b)(a + 2 - 3b)$

(b) $(a^2 + 2ab + b^2) - 4$

1. Recognize that $a^2 + 2ab + b^2$ can be factored as $(a + b)^2$.
2. So, the expression becomes $(a + b)^2 - 2^2$.
3. Again, this is a difference of squares: $(a + b + 2)(a + b - 2)$.

Simplified Form:

$(a + b + 2)(a + b - 2)$

(c) $16 - (a^2 - 6a + 9)$

1. Rewrite $a^2 - 6a + 9$ as $(a - 3)^2$.
2. The expression now is $4^2 - (a - 3)^2$.
3. This is also a difference of squares: $(4 + (a - 3))(4 - (a - 3))$.
4. Simplifying further, we get $(a + 1)(7 - a)$.

Simplified Form:

$(a + 1)(7 - a)$

(d) $4a^2 + (-b^2 + 2bc - c^2)$

1. Observe that $-b^2 + 2bc - c^2$ is equivalent to $-(b^2 - 2bc + c^2)$, which factors as $-(b - c)^2$.
2. Thus, the expression becomes $4a^2 - (b - c)^2$.
3. Using the difference of squares: $(2a + (b - c))(2a - (b - c))$.

Simplified Form:

$(2a + b - c)(2a - b + c)$

Would you like further clarification on any step, or details on specific parts of the simplifications?

Related Questions

1. How do you factor a trinomial like $a^2 + 4a + 4$?
2. What is the difference of squares formula, and how is it used in factoring?
3. How can we recognize expressions that can be factored as perfect squares?
4. What are the conditions for a quadratic expression to be factored as a difference of squares?
5. How can factoring be used to solve polynomial equations?

Tip: When factoring, always look for common patterns like perfect squares and differences of squares; these can simplify expressions quickly.