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Let's break down the factorization of the two expressions.

### (i) \( (x + 2)^2 + p^2 + 2p(x + 2) \)
1. First, expand the terms:
   - \( (x + 2)^2 = x^2 + 4x + 4 \)
   - \( 2p(x + 2) = 2px + 4p \)

So the expression becomes:
\[
x^2 + 4x + 4 + p^2 + 2px + 4p
\]
Group like terms:
\[
x^2 + (4x + 2px) + p^2 + 4 + 4p
\]
Factor out common terms:
\[
x^2 + x(4 + 2p) + (p^2 + 4p + 4)
\]
Notice that \( p^2 + 4p + 4 \) is a perfect square trinomial:
\[
p^2 + 4p + 4 = (p + 2)^2
\]
Thus, the expression becomes:
\[
x^2 + x(4 + 2p) + (p + 2)^2
\]
This is the factorized form of the first expression.

---

### (ii) \( m^8 - 11m^4n^4 - 80n^8 \)
This is a quadratic in terms of \( m^4 \) and \( n^4 \), so we can treat \( m^4 \) as one variable and \( n^4 \) as another.

Rewrite the expression as:
\[
m^8 - 11m^4n^4 - 80n^8 = (m^4)^2 - 11m^4n^4 - 80(n^4)^2
\]

We can factor this using the quadratic method:
We are looking for two numbers that multiply to \( -80 \times 1 = -80 \) and add to \( -11 \). The factors are \( -16 \) and \( 5 \).

Thus, we can factor it as:
\[
(m^4 - 16n^4)(m^4 + 5n^4)
\]

Now, factor further:
- \( m^4 - 16n^4 \) is a difference of squares:
\[
m^4 - 16n^4 = (m^2 - 4n^2)(m^2 + 4n^2)
\]

Therefore, the fully factorized form is:
\[
(m^2 - 4n^2)(m^2 + 4n^2)(m^4 + 5n^4)
\]
Finally, recognize that \( m^2 - 4n^2 \) is also a difference of squares:
\[
m^2 - 4n^2 = (m - 2n)(m + 2n)
\]

Thus, the completely factorized form is:
\[
(m - 2n)(m + 2n)(m^2 + 4n^2)(m^4 + 5n^4)
\]

---

Let me know if you'd like any details about the steps, or if you have more questions! Here are 5 related questions:

1. How do you identify a perfect square trinomial?
2. What is the general formula for the difference of squares?
3. Can every quadratic-like expression be factorized? Why or why not?
4. How would you factor a sum of cubes, such as \( a^3 + b^3 \)?
5. What is the significance of recognizing common factor patterns in polynomials?

**Tip:** Always check if a polynomial is a special type, like a difference of squares or a perfect square trinomial, before trying complex factoring techniques!

Factorise: (i) (x + 2)^2 + p^2 + 2p(x + 2), (ii) m^8 - 11m^4n^4 - 80n^8

Learn how to factorize expressions like (x + 2)^2 + p^2 + 2p(x + 2) and m^8 - 11m^4n^4 - 80n^8. This problem involves key algebraic concepts such as perfect square trinomials, difference of squares, and polynomial factorization techniques. Suitable for students in Grades 10-12.

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