Let's solve the given question: Factorize the expressions.

(i) $ax^2 + bx$

Factorize by taking the common factor: $x(ax + b)$ Answer: $x(ax + b)$

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

Factorize by taking the greatest common factor (GCF), which is $7$: $7(p^2 + 3q^2)$ Answer: $7(p^2 + 3q^2)$

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

Take the GCF, $2x$, from all terms: $2x(x^2 + y^2 + z^2)$ Answer: $2x(x^2 + y^2 + z^2)$

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

Group the terms strategically: $(am^2 + an^2) + (bm^2 + bn^2)$ Factorize each group: $a(m^2 + n^2) + b(m^2 + n^2)$ Take $(m^2 + n^2)$ as the common factor: $(m^2 + n^2)(a + b)$ Answer: $(m^2 + n^2)(a + b)$

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

Group and take common factors from both groups: $l(m + 1) + (m + 1)$ Now take $(m + 1)$ as the common factor: $(m + 1)(l + 1)$ Answer: $(m + 1)(l + 1)$

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

Factorize $(y + z)$: $(y + z)(y + 9)$ Answer: $(y + z)(y + 9)$

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

Group and rearrange the terms: $(5y^2 - 20y) + (-8z + 2yz)$ Factorize each group: $5y(y - 4) + (-2z)(4 - y)$ Simplify by factoring $(y - 4)$: $(5y - 2z)(y - 4)$ Answer: $(5y - 2z)(y - 4)$

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

Group and rearrange terms: $(10ab + 5b) + (4a + 2)$ Factorize each group: $5b(2a + 1) + 2(2a + 1)$ Simplify by factoring $(2a + 1)$: $(5b + 2)(2a + 1)$ Answer: $(5b + 2)(2a + 1)$

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

Group and rearrange terms: $(6xy - 9x) + (-4y + 6)$ Factorize each group: $3x(2y - 3) - 2(2y - 3)$ Simplify by factoring $(2y - 3)$: $(3x - 2)(2y - 3)$ Answer: $(3x - 2)(2y - 3)$

Let me know if you'd like further explanation for any specific part!

Related Questions:

1. How can we identify the greatest common factor (GCF) when factorizing?
2. What is the difference between grouping and taking out common terms?
3. Can all polynomials be factorized? Why or why not?
4. How do you factorize quadratics of the form $ax^2 + bx + c$?
5. What are some special cases of polynomials (e.g., perfect squares, difference of squares)?

Tip:

When grouping terms for factorization, always look for common factors and strategically rearrange terms to simplify the process.