1. Factorise. (a) \( a^{2}-4 a b+4 b^{2}-16 x^{2} \) (b) \( c^{2}-a^{2}+2 a b-b^{2} \) (c) \( a^{2}+2 a b+b^{2}-9 x^{2} \) (d) \( a^{2}-2 x y-y^{2}-x^{2} \) e) \( 16 x^{4}-a^{2}+6 a b-9 b^{2} \) (f) \( 25 y^{4}-1-9 m^{2}+6 m \) (g) \( 9 r^{2}-4+12 a-9 a^{2} \) (h) \( 1-81 a^{2}-18 a b-b^{2} \) i) \( 20 a b+4 a^{2}+25 b^{2}-9 x^{4} \) j) \( 24 x y-3 z^{2}+3 y^{2}+48 x^{2} \)

We will factorise each expression step‐by‐step. ───────────────────────────── Example (a) Factorise   a² – 4ab + 4b² – 16x² Step 1. Notice that the first three terms form a perfect square:   a² – 4ab + 4b² = (a – 2b)² Step 2. Then your expression becomes a difference of two squares:   (a – 2b)² – (4x)² Step 3. Factor using the difference‐of‐squares formula:   X² – Y² = (X – Y)(X + Y) Thus,   (a – 2b – 4x)(a – 2b + 4x) ───────────────────────────── Example (b) Factorise   c² – a² + 2ab – b² Step 1. Group the last three terms:   a² – 2ab + b² = (a – b)² So rewrite the expression as:   c² – (a – b)² Step 2. Again a difference of two squares:   = (c – (a – b)) (c + (a – b)) which simplifies to   = (c – a + b)(c + a – b) ───────────────────────────── Example (c) Factorise   a² + 2ab + b² – 9x² Step 1. Recognize a² + 2ab + b² = (a + b)², so:   = (a + b)² – (3x)² Step 2. Factor the difference of two squares:   = (a + b – 3x)(a + b + 3x) ───────────────────────────── Example (d) Factorise   a² – 2xy – y² – x² Step 1. Notice that the last three terms can be regrouped:   x² + 2xy + y² = (x + y)² Thus, write:   a² – (x + y)² Step 2. Factor as a difference of squares:   = (a – (x + y))(a + (x + y))   = (a – x – y)(a + x + y) ───────────────────────────── Example (e) Factorise   16x⁴ – a² + 6ab – 9b² Step 1. Group the last three terms. Note that:   a² – 6ab + 9b² = (a – 3b)² So write:   16x⁴ – (a – 3b)² Step 2. Recognise 16x⁴ = (4x²)², then use difference of two squares:   = (4x² – (a – 3b)) (4x² + (a – 3b))   = (4x² – a + 3b)(4x² + a – 3b) ───────────────────────────── Example (f) Factorise   25y⁴ – 1 – 9m² + 6m Step 1. Rearrange the terms to group two pairs:   (25y⁴ – 1) – (9m² – 6m) Step 2. Notice that 25y⁴ – 1 is a difference of two squares,   25y⁴ – 1 = (5y²)² – 1² = (5y² – 1)(5y² + 1)   However, the second group does not factor nicely by itself. A better idea is to rewrite the last three terms as a complete square:   9m² – 6m + 1 = (3m – 1)²  so that   –(9m² – 6m + 1) = –(3m – 1)² Step 3. Now write the whole expression as:   25y⁴ – (3m – 1)² Step 4. Then factor as a difference of squares:   = (5y² – (3m – 1)) (5y² + (3m – 1))   = (5y² – 3m + 1)(5y² + 3m – 1) ───────────────────────────── Example (g) Factorise   9r² – 4 + 12a – 9a² Step 1. Rearrange the terms to group those in a:   9r² – 9a² + 12a – 4 Step 2. Notice that 9a² – 12a + 4 = (3a – 2)². Thus write:   9r² – (3a – 2)² Step 3. Factor as a difference of two squares:   = (3r – (3a – 2)) (3r + (3a – 2))   = (3r – 3a + 2)(3r + 3a – 2) ───────────────────────────── Example (h) Factorise   1 – 81a² – 18ab – b² Step 1. Group the last three terms. Notice that:   81a² + 18ab + b² = (9a + b)² Step 2. Then the expression becomes:   1 – (9a + b)² Step 3. Factor as a difference of two squares:   = (1 – (9a + b))(1 + (9a + b))   = (1 – 9a – b)(1 + 9a + b) ───────────────────────────── Example (i) Factorise   20ab + 4a² + 25b² – 9x⁴ Step 1. Rearrange the terms so that the a–b part is together:   4a² + 20ab + 25b² = (2a + 5b)² Step 2. Then the expression becomes:   (2a + 5b)² – 9x⁴   and 9x⁴ = (3x²)² Step 3. Factor by difference of squares:   = (2a + 5b – 3x²)(2a + 5b + 3x²) ───────────────────────────── Example (j) Factorise   24xy – 3z² + 3y² + 48x² Step 1. Group the terms in x and y:   48x² + 24xy + 3y² Notice that this is a perfect square since   48x² + 24xy + 3y² = 3(16x² + 8xy + y²) = 3(4x + y)² Step 2. Now the whole expression becomes:   3(4x + y)² – 3z² Step 3. Factor out the common factor 3 and use the difference of squares:   = 3[(4x + y)² – z²]   = 3[(4x + y) – z][(4x + y) + z]   = 3(4x + y – z)(4x + y + z) ───────────────────────────── Summary of Answers (a) (a – 2b – 4x)(a – 2b + 4x) (b) (c – a + b)(c + a – b) (c) (a + b – 3x)(a + b + 3x) (d) (a – x – y)(a + x + y) (e) (4x² – a + 3b)(4x² + a – 3b) (f) (5y² – 3m + 1)(5y² + 3m – 1)