The problem requires factorizing the given algebraic expressions. Let’s work through each part step by step:

(i) $ax^2 + bx$

Solution:
Factor out the common term $x$:
$ax^2 + bx = x(ax + b)$

(ii) $7p^2 + 21q^2$

Solution:
Factor out the greatest common factor (GCF), which is $7$:
$7p^2 + 21q^2 = 7(p^2 + 3q^2)$

(iii) $2x^3 + 2xy^2 + 2xz^2$

Solution:
Factor out the GCF, $2x$:
$2x^3 + 2xy^2 + 2xz^2 = 2x(x^2 + y^2 + z^2)$

(iv) $am^2 + bm^2 + bn^2 + an^2$

Solution:
Group terms strategically:
$(am^2 + bm^2) + (bn^2 + an^2)$
Factor out common terms from each group:
$m^2(a + b) + n^2(b + a)$
Notice that $(a + b) = (b + a)$, so factor it out:
$(a + b)(m^2 + n^2)$

(v) $(lm + l) + (m + 1)$

Solution:
Group terms and factor common terms from each group:
$l(m + 1) + (m + 1)$
Now, factor out $(m + 1)$:
$(m + 1)(l + 1)$

(vi) $y(y + z) + 9(y + z)$

Solution:
Factor out the common term $(y + z)$:
$y(y + z) + 9(y + z) = (y + z)(y + 9)$

(vii) $5y^2 - 20yz + 8z$

Solution:
Group terms:
$5y^2 - 20yz + 8z = 5y^2 - 20yz + 8z = (5y^2 - 20yz) + 8z$
Factor common terms for simplification:
Work it fully for compact.Let’s complete the solution for (vii) and continue to the rest:

(vii) $5y^2 - 20y + 8z$

Solution:
Group the terms:
$(5y^2 - 20y) + 8z$
Factor $5y$ from the first group:
$5y(y - 4) + 8z$
This expression cannot be factored further unless more grouping or relationships are evident.

(viii) $10ab + 4a + 5b + 2$

Solution:
Group terms:
$(10ab + 5b) + (4a + 2)$
Factor $5b$ from the first group and $2$ from the second group:
$5b(2a + 1) + 2(2a + 1)$
Now factor $(2a + 1)$:
$(2a + 1)(5b + 2)$

(ix) $6xy - 4y + 6 - 9x$

Solution:
Group terms:
$(6xy - 4y) + (6 - 9x)$
Factor $2y$ from the first group and $3$ from the second group:
$2y(3x - 2) - 3(3x - 2)$
Now factor $(3x - 2)$:
$(3x - 2)(2y - 3)$

Let me know if you’d like detailed steps or explanations for any specific part!

Related Questions:

1. How do you determine the greatest common factor (GCF) in algebraic expressions?
2. What are some tips for grouping terms to simplify factorization?
3. How do you factorize cubic or higher-degree polynomials effectively?
4. Can all expressions be factorized? Why or why not?
5. What are common factorization mistakes to avoid?

Tip:

Always look for common factors first before attempting complex grouping or advanced methods!