Students can practice their algebra skills with this worksheet covering simplification and the difference of two squares.

Here are the solutions to the problems on the worksheet. I have worked through each one step-by-step to ensure accuracy.

Section A: Simplify then factorise

1) $x^2 - 6x - 2x + 12$
* Simplify: Combine the middle terms ($-6x - 2x = -8x$).
Expression becomes: $x^2 - 8x + 12$
* Factorise: Find two numbers that multiply to $+12$ and add to $-8$. The numbers are $-2$ and $-6$.
* Result: $(x - 2)(x - 6)$

2) $d(d - 5) - 84$
* Simplify: Expand the bracket ($d \times d = d^2$, $d \times -5 = -5d$).
Expression becomes: $d^2 - 5d - 84$
* Factorise: Find two numbers that multiply to $-84$ and add to $-5$. The numbers are $-12$ and $+7$.
* Result: $(d - 12)(d + 7)$

3) $b^2 + 2(b - 4)$
* Simplify: Expand the bracket ($2 \times b = 2b$, $2 \times -4 = -8$).
Expression becomes: $b^2 + 2b - 8$
* Factorise: Find two numbers that multiply to $-8$ and add to $+2$. The numbers are $+4$ and $-2$.
* Result: $(b + 4)(b - 2)$

4) $x^2 - 3(2x + 9)$
* Simplify: Expand the bracket ($-3 \times 2x = -6x$, $-3 \times 9 = -27$).
Expression becomes: $x^2 - 6x - 27$
* Factorise: Find two numbers that multiply to $-27$ and add to $-6$. The numbers are $-9$ and $+3$.
* Result: $(x - 9)(x + 3)$

5) $c(c + 8) - 48$
* Simplify: Expand the bracket ($c \times c = c^2$, $c \times 8 = 8c$).
Expression becomes: $c^2 + 8c - 48$
* Factorise: Find two numbers that multiply to $-48$ and add to $+8$. The numbers are $+12$ and $-4$.
* Result: $(c + 12)(c - 4)$

6) $3a(a - 2) - 4a + 3$
* Simplify: Expand the first part ($3a^2 - 6a$). Then combine with the rest: $3a^2 - 6a - 4a + 3$.
Combine like terms ($-6a - 4a = -10a$): $3a^2 - 10a + 3$
* Factorise: We need factors of $3 \times 3 = 9$ that add to $-10$. These are $-9$ and $-1$.
Split the middle term: $3a^2 - 9a - a + 3$.
Group: $3a(a - 3) - 1(a - 3)$.
* Result: $(3a - 1)(a - 3)$

7) $5w(w - 2) - 4w - 3$
* Simplify: Expand ($5w^2 - 10w$). Combine: $5w^2 - 10w - 4w - 3$.
Combine like terms ($-10w - 4w = -14w$): $5w^2 - 14w - 3$
* Factorise: Factors of $5 \times -3 = -15$ that add to $-14$. These are $-15$ and $+1$.
Split middle term: $5w^2 - 15w + w - 3$.
Group: $5w(w - 3) + 1(w - 3)$.
* Result: $(5w + 1)(w - 3)$

8) $3(6 - 5s) + s^2 + s^2$
* Simplify: Expand ($18 - 15s$). Combine $s^2$ terms ($s^2 + s^2 = 2s^2$).
Rearrange into standard form: $2s^2 - 15s + 18$
* Factorise: Factors of $2 \times 18 = 36$ that add to $-15$. These are $-12$ and $-3$.
Split middle term: $2s^2 - 12s - 3s + 18$.
Group: $2s(s - 6) - 3(s - 6)$.
* Result: $(2s - 3)(s - 6)$

9) $3 + 2y(4y + 5)$
* Simplify: Expand ($8y^2 + 10y$). Add the 3.
Expression becomes: $8y^2 + 10y + 3$
* Factorise: Factors of $8 \times 3 = 24$ that add to $10$. These are $6$ and $4$.
Split middle term: $8y^2 + 4y + 6y + 3$.
Group: $4y(2y + 1) + 3(2y + 1)$.
* Result: $(4y + 3)(2y + 1)$

10) $9x^2 - (x - 3)^2$
* Simplify: Expand the squared bracket carefully: $(x-3)(x-3) = x^2 - 6x + 9$.
Subtract this from $9x^2$: $9x^2 - (x^2 - 6x + 9) = 9x^2 - x^2 + 6x - 9$.
Combine like terms: $8x^2 + 6x - 9$
* Factorise: Factors of $8 \times -9 = -72$ that add to $6$. These are $12$ and $-6$.
Split middle term: $8x^2 + 12x - 6x - 9$.
Group: $4x(2x + 3) - 3(2x + 3)$.
* Result: $(4x - 3)(2x + 3)$

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Section B: Factorise (Difference of Two Squares)

*Rule: $a^2 - b^2 = (a+b)(a-b)$*

1) $x^2 - 4 = (x + 2)(x - 2)$
2) $s^2 - 25 = (s + 5)(s - 5)$
3) $t^2 - 64 = (t + 8)(t - 8)$
4) $9 - y^2 = (3 + y)(3 - y)$
5) $49 - p^2 = (7 + p)(7 - p)$
6) $4q^2 - 121 = (2q + 11)(2q - 11)$
7) $81 - 25k^2 = (9 + 5k)(9 - 5k)$
8) $1 - 400d^2 = (1 + 20d)(1 - 20d)$
9) $600v^2 - 6 = 6(100v^2 - 1) = 6(10v + 1)(10v - 1)$ *(Take out common factor 6 first)*
10) $a^2 - b^2 = (a + b)(a - b)$
11) $x^2 - 9y^2 = (x + 3y)(x - 3y)$
12) $4c^2 - d^2 = (2c + d)(2c - d)$
13) $16s^2 - 9t^2 = (4s + 3t)(4s - 3t)$
14) $49w^2 - 100v^2 = (7w + 10v)(7w - 10v)$
15) $32p^2 - 18q^2 = 2(16p^2 - 9q^2) = 2(4p + 3q)(4p - 3q)$ *(Take out common factor 2 first)*
16) $48x^2 - 12y^2 = 12(4x^2 - y^2) = 12(2x + y)(2x - y)$ *(Take out common factor 12 first)*
17) $45a^2 - 125b^2 = 5(9a^2 - 25b^2) = 5(3a + 5b)(3a - 5b)$ *(Take out common factor 5 first)*
18) $72x^2 - 242y^2 = 2(36x^2 - 121y^2) = 2(6x + 11y)(6x - 11y)$ *(Take out common factor 2 first)*
19) $a^2b^2 - c^2 = (ab + c)(ab - c)$
20) $9s - 4s^3 = s(9 - 4s^2) = s(3 + 2s)(3 - 2s)$ *(Take out common factor s first)*
21) $(xy)^2 - 4z^2 = (xy + 2z)(xy - 2z)$
22) $64t^4 - 16s^4 = 16(4t^4 - s^4) = 16((2t^2)^2 - (s^2)^2) = 16(2t^2 + s^2)(2t^2 - s^2)$
23) $(4x^2)^2 - 36y^2 = 16x^4 - 36y^2$. Take out 4: $4(4x^4 - 9y^2) = 4((2x^2)^2 - (3y)^2) = 4(2x^2 + 3y)(2x^2 - 3y)$
24) $27a^4 - 12b^2 = 3(9a^4 - 4b^2) = 3((3a^2)^2 - (2b)^2) = 3(3a^2 + 2b)(3a^2 - 2b)$

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Extension: Difference of Two Squares

1) $4x^2 - (x - 2)^2$
This is in the form $A^2 - B^2$, where $A = 2x$ and $B = (x - 2)$.
Formula: $(A + B)(A - B)$
* First bracket: $2x + (x - 2) = 3x - 2$
* Second bracket: $2x - (x - 2) = 2x - x + 2 = x + 2$
* Result: $(3x - 2)(x + 2)$

2) $(2x + 1)^2 - (x - 4)^2$
This is in the form $A^2 - B^2$, where $A = (2x + 1)$ and $B = (x - 4)$.
Formula: $(A + B)(A - B)$
* First bracket: $(2x + 1) + (x - 4) = 3x - 3$
*Note: We can factor out a 3 here to get $3(x-1)$.*
* Second bracket: $(2x + 1) - (x - 4) = 2x + 1 - x + 4 = x + 5$
* Combine: $3(x - 1)(x + 5)$
* Result: $3(x - 1)(x + 5)$

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Final Answer:
Section A
1) $(x - 2)(x - 6)$
2) $(d - 12)(d + 7)$
3) $(b + 4)(b - 2)$
4) $(x - 9)(x + 3)$
5) $(c + 12)(c - 4)$
6) $(3a - 1)(a - 3)$
7) $(5w + 1)(w - 3)$
8) $(2s - 3)(s - 6)$
9) $(4y + 3)(2y + 1)$
10) $(4x - 3)(2x + 3)$

Section B
1) $(x + 2)(x - 2)$
2) $(s + 5)(s - 5)$
3) $(t + 8)(t - 8)$
4) $(3 + y)(3 - y)$
5) $(7 + p)(7 - p)$
6) $(2q + 11)(2q - 11)$
7) $(9 + 5k)(9 - 5k)$
8) $(1 + 20d)(1 - 20d)$
9) $6(10v + 1)(10v - 1)$
10) $(a + b)(a - b)$
11) $(x + 3y)(x - 3y)$
12) $(2c + d)(2c - d)$
13) $(4s + 3t)(4s - 3t)$
14) $(7w + 10v)(7w - 10v)$
15) $2(4p + 3q)(4p - 3q)$
16) $12(2x + y)(2x - y)$
17) $5(3a + 5b)(3a - 5b)$
18) $2(6x + 11y)(6x - 11y)$
19) $(ab + c)(ab - c)$
20) $s(3 + 2s)(3 - 2s)$
21) $(xy + 2z)(xy - 2z)$
22) $16(2t^2 + s^2)(2t^2 - s^2)$
23) $4(2x^2 + 3y)(2x^2 - 3y)$
24) $3(3a^2 + 2b)(3a^2 - 2b)$

Extension
1) $(3x - 2)(x + 2)$
2) $3(x - 1)(x + 5)$