I see you've uploaded a set of math problems. I'll start by helping with some of the questions listed. Here's a breakdown of solutions:

Short Answer Questions

1. Subtract $5a^2b^2c^2$ from $7a^2b^2c^2$:

   $7a^2b^2c^2 - 5a^2b^2c^2 = 2a^2b^2c^2$

2. Simplify $(ab - c)^2 + 2abc$:

   First, expand $(ab - c)^2$:

   $(ab - c)^2 = a^2b^2 - 2abc + c^2$

   Now, add $2abc$:

   $a^2b^2 - 2abc + c^2 + 2abc = a^2b^2 + c^2$

3. Expand using suitable identities:

   $\left( \frac{4a}{5} + \frac{5b}{4} \right)^2$

   Use the identity $(x + y)^2 = x^2 + 2xy + y^2$.

   $= \left( \frac{4a}{5} \right)^2 + 2\left( \frac{4a}{5} \times \frac{5b}{4} \right) + \left( \frac{5b}{4} \right)^2$

   Simplifying each term:

   = \frac{16a^2}{25} + 2ab + \frac{25b^2}{16}$$ Hence, the expanded form is: $$\frac{16a^2}{25} + 2ab + \frac{25b^2}{16}$$

4. Carry out the division:

   $\frac{76x^3y^2}{19x^2y}$

   Simplify the coefficients and the powers of $x$ and $y$:

   $\frac{76}{19} \times \frac{x^3}{x^2} \times \frac{y^2}{y} = 4x \cdot y = 4xy$

5. Factorize $x^2 - 8x + 16$:

   Recognize this as a perfect square trinomial:

   $x^2 - 8x + 16 = (x - 4)^2$

Let me know if you need more details or help with the remaining questions.

Here are 5 follow-up questions related to these topics:

1. How do you recognize a perfect square trinomial?
2. What are the general steps to simplify rational expressions like the division in question 4?
3. How do you identify which factoring method to use for different quadratic expressions?
4. Can you explain how the identity $(a + b)^2 = a^2 + 2ab + b^2$ is derived?
5. How do you handle negative exponents during simplification?

Tip: When factorizing quadratic expressions, always check if it’s a perfect square trinomial or if it can be factored by grouping.