Let me analyze the problems in the image and solve them step by step. I'll address each question comprehensively.

3. Fully Factorized Form by Removing the HCF

a) $4a + 12b$

• $HCF = 4$
• $4a + 12b = 4(a + 3b)$

b) $6x - 9x^2$

• $HCF = 3x$
• $6x - 9x^2 = 3x(2 - 3x)$

c) $-5x^2y - 10xy$

• $HCF = -5xy$
• $-5x^2y - 10xy = -5xy(x + 2)$

d) $3(x - 7) + x(x - 7)$

• $Common\ factor: (x - 7)$
• $3(x - 7) + x(x - 7) = (x - 7)(3 + x)$

e) $x(2x + 1) - (2x + 1)$

• $Common\ factor: (2x + 1)$
• $x(2x + 1) - (2x + 1) = (2x + 1)(x - 1)$

f) $(x - 2)^2 - 4(x - 2)$

• $Common\ factor: (x - 2)$
• $(x - 2)^2 - 4(x - 2) = (x - 2)[(x - 2) - 4] = (x - 2)(x - 6)$

4. Difference of Two Squares

a) $x^2 - 100$

• $x^2 - 100 = (x - 10)(x + 10)$

b) $3x^2 - 48$

• $Factor\ out\ 3$: $3(x^2 - 16)$
• $x^2 - 16 = (x - 4)(x + 4)$
• Final: $3(x - 4)(x + 4)$

c) $25x^2 - y^2$

• $25x^2 - y^2 = (5x - y)(5x + y)$

d) $49 - 9x^2$

• $49 - 9x^2 = (7 - 3x)(7 + 3x)$

e) $(x - 3)^2 - 81$

• $(x - 3)^2 - 81 = [(x - 3) - 9][(x - 3) + 9]$
• Final: $(x - 12)(x + 6)$

f) $1 - x^2$

• $1 - x^2 = (1 - x)(1 + x)$

5. Factorize by Grouping in Pairs

a) $x^2 - 3x + 6x - 18$

• Group as $(x^2 - 3x) + (6x - 18)$
• $x(x - 3) + 6(x - 3)$
• Final: $(x + 6)(x - 3)$

b) $4x^2 + 10x - 2x - 5$

• Group as $(4x^2 + 10x) - (2x + 5)$
• $2x(2x + 5) - 1(2x + 5)$
• Final: $(2x - 1)(2x + 5)$

c) $3x - 8b + 2bx - 12$

• Group as $(3x - 8b) + (2bx - 12)$
• $x(3 - 8b) + 2b(x - 6)$
• Cannot group further meaningfully here.

6. Factorize Trinomials

a) $x^2 + 8x + 15$

• $Factors\ of\ 15:\ 3,\ 5$
• Final: $(x + 3)(x + 5)$

b) $x^2 - 3x - 18$

• $Factors\ of\ -18:\ -6,\ 3$
• Final: $(x - 6)(x + 3)$

c) $x^2 - 7x + 6$

• $Factors\ of\ 6:\ -6,\ -1$
• Final: $(x - 6)(x - 1)$

d) $3x^2 + 15x - 42$

• Factor out 3: $3(x^2 + 5x - 14)$
• $Factors\ of\ -14:\ 7,\ -2$
• Final: $3(x + 7)(x - 2)$

e) $2x^2 + 16x + 32$

• Factor out 2: $2(x^2 + 8x + 16)$
• $x^2 + 8x + 16 = (x + 4)^2$
• Final: $2(x + 4)^2$

f) $5x^2 + 17x + 6$

• $Factors\ of\ 30:\ 15,\ 2$
• $5x^2 + 15x + 2x + 6$
• Final: $(5x + 2)(x + 3)$

g) $4x^2 - 4x - 3$

• $Factors\ of\ -12:\ -6,\ 2$
• $4x^2 - 6x + 2x - 3$
• Final: $(2x - 3)(2x + 1)$

h) $6x^2 - 17x + 12$

• $Factors\ of\ 72:\ -8,\ -9$
• $6x^2 - 9x - 8x + 12$
• Final: $(3x - 4)(2x - 3)$

Would you like a detailed explanation for any of these steps? Let me know!

Related Questions

1. What is the general process for factoring out the greatest common factor (GCF)?
2. How do you identify if an expression is a difference of two squares?
3. What are some strategies for factoring by grouping effectively?
4. How can trinomial expressions be solved systematically?
5. What role does the distributive property play in factorization?

Tip

Always check your factorized results by expanding them back to ensure accuracy!