The task requires simplifying each expression by expanding and collecting like terms. Here's the step-by-step solution for each:

a) $(x - 4)^2$
Use the formula $(a - b)^2 = a^2 - 2ab + b^2$:
$(x - 4)^2 = x^2 - 2(4)(x) + 4^2 = x^2 - 8x + 16$
Final Answer: $x^2 - 8x + 16$

b) $(2x - 3)^2$
Use the same formula $(a - b)^2 = a^2 - 2ab + b^2$:
$(2x - 3)^2 = (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9$
Final Answer: $4x^2 - 12x + 9$

c) $(6 - 2x)^2$
Apply $(a - b)^2 = a^2 - 2ab + b^2$:
$(6 - 2x)^2 = 6^2 - 2(6)(2x) + (2x)^2 = 36 - 24x + 4x^2$
Rearrange terms:
$4x^2 - 24x + 36$
Final Answer: $4x^2 - 24x + 36$

d) $(x - \frac{1}{2})^2$
Expand using $(a - b)^2 = a^2 - 2ab + b^2$:
$(x - \frac{1}{2})^2 = x^2 - 2(x)(\frac{1}{2}) + (\frac{1}{2})^2 = x^2 - x + \frac{1}{4}$
Final Answer: $x^2 - x + \frac{1}{4}$

e) $(x - \sqrt{5})^2$
Expand using the same formula:
$(x - \sqrt{5})^2 = x^2 - 2(x)(\sqrt{5}) + (\sqrt{5})^2 = x^2 - 2\sqrt{5}x + 5$
Final Answer: $x^2 - 2\sqrt{5}x + 5$

f) $(x - 2\sqrt{3})^2$
Apply the formula:
$(x - 2\sqrt{3})^2 = x^2 - 2(x)(2\sqrt{3}) + (2\sqrt{3})^2 = x^2 - 4\sqrt{3}x + 12$
Final Answer: $x^2 - 4\sqrt{3}x + 12$

Would you like me to break down any step in more detail?

Here are 5 related questions to deepen understanding:

1. What is the difference between expanding and factoring expressions?
2. How do you recognize and use special products like $(a - b)^2$ efficiently?
3. How do square roots behave in algebraic expansions?
4. What is the effect of coefficients (like $2x$ or $6$) in these expansions?
5. How can these expansions be applied in solving quadratic equations?

Tip: Always double-check your expansions by re-factoring the result to ensure correctness.