Simplify. (i) `(x^2-5) (x+5)+25` (ii) `(a^2+5) (b^3+3)+5` (iii) `(t+s^2) (t^2-s)` (iv) `(a+b)(c-d)+(9a-b)(c+d)+2(ac+bd)` (v) `(x+y)(2x+y)+(x+2y)+(x+2y)(x-y)` (vi) `(x+y)(x^2-xy+y^2)` (vii) `(1.5x-4y)(1.5x+4y+3)-4.5x+12y` (viii) `(a+b+c)(a+b-c)`

Explanation

Let's simplify each part step by step:

### (i) Simplify \( (x^2 - 5)(x + 5) + 25 \)

1. **Distribute \( (x^2 - 5) \) with \( (x + 5) \)**:
\[ (x^2 - 5)(x + 5) = x^2 \cdot x + x^2 \cdot 5 - 5 \cdot x - 5 \cdot 5 = x^3 + 5x^2 - 5x - 25 \]

2. **Add 25**:
\[ x^3 + 5x^2 - 5x - 25 + 25 = x^3 + 5x^2 - 5x \]

**Final Answer**: \( x^3 + 5x^2 - 5x \)

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### (ii) Simplify \( (a^2 + 5)(b^3 + 3) + 5 \)

1. **Distribute \( (a^2 + 5) \) with \( (b^3 + 3) \)**:
\[ (a^2 + 5)(b^3 + 3) = a^2 \cdot b^3 + a^2 \cdot 3 + 5 \cdot b^3 + 5 \cdot 3 = a^2b^3 + 3a^2 + 5b^3 + 15 \]

2. **Add 5**:
\[ a^2b^3 + 3a^2 + 5b^3 + 15 + 5 = a^2b^3 + 3a^2 + 5b^3 + 20 \]

**Final Answer**: \( a^2b^3 + 3a^2 + 5b^3 + 20 \)

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### (iii) Simplify \( (t + s^2)(t^2 - s) \)

1. **Distribute \( (t + s^2) \) with \( (t^2 - s) \)**:
\[ (t + s^2)(t^2 - s) = t \cdot t^2 - t \cdot s + s^2 \cdot t^2 - s^2 \cdot s = t^3 - ts + s^2t^2 - s^3 \]

**Final Answer**: \( t^3 - ts + s^2t^2 - s^3 \)

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### (iv) Simplify \( (a + b)(c - d) + (9a - b)(c + d) + 2(ac + bd) \)

1. **Distribute each term**:
\[ (a + b)(c - d) = ac - ad + bc - bd \]
\[ (9a - b)(c + d) = 9ac + 9ad - bc - bd \]
\[ 2(ac + bd) = 2ac + 2bd \]

2. **Combine all parts**:
\[ (ac - ad + bc - bd) + (9ac + 9ad - bc - bd) + (2ac + 2bd) \]
Combine like terms:
\[ (ac + 9ac + 2ac) + (-ad + 9ad) + (bc - bc) + (-bd - bd + 2bd) = 12ac + 8ad - 2bd \]

**Final Answer**: \( 12ac + 8ad - 2bd \)

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### (v) Simplify \( (x + y)(2x + y) + (x + 2y) + (x + 2y)(x - y) \)

1. **Distribute each term**:
\[ (x + y)(2x + y) = 2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2 \]
\[ (x + 2y)(x - y) = x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2 \]

2. **Combine all parts**:
\[ (2x^2 + 3xy + y^2) + (x + 2y) + (x^2 + xy - 2y^2) \]
Combine like terms:
\[ (2x^2 + x^2) + (3xy + xy) + (y^2 - 2y^2) + x + 2y = 3x^2 + 4xy - y^2 + x + 2y \]

**Final Answer**: \( 3x^2 + 4xy - y^2 + x + 2y \)

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### (vi) Simplify \( (x + y)(x^2 - xy + y^2) \)

1. **Distribute \( (x + y) \)**:
\[ (x + y)(x^2 - xy + y^2) = x \cdot x^2 - x \cdot xy + x \cdot y^2 + y \cdot x^2 - y \cdot xy + y \cdot y^2 \]
Combine like terms:
\[ x^3 - x^2y + xy^2 + yx^2 - y^2x + y^3 = x^3 + y^3 \]

**Final Answer**: \( x^3 + y^3 \)

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### (vii) Simplify \( (1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y \)

1. **Distribute \( (1.5x - 4y) \)**:
\[ (1.5x - 4y)(1.5x + 4y + 3) = 1.5x \cdot 1.5x + 1.5x \cdot 4y + 1.5x \cdot 3 - 4y \cdot 1.5x - 4y \cdot 4y - 4y \cdot 3 \]
Simplifying gives:
\[ 2.25x^2 + 6xy + 4.5x - 6xy - 16y^2 - 12y \]

2. **Combine with \( -4.5x + 12y \)**:
\[ 2.25x^2 - 16y^2 + (4.5x - 4.5x) + (-12y + 12y) = 2.25x^2 - 16y^2 \]

**Final Answer**: \( 2.25x^2 - 16y^2 \)

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### (viii) Simplify \( (a + b + c)(a + b - c) \)

1. **Distribute \( (a + b + c) \)**:
\[ (a + b + c)(a + b - c) = a(a + b - c) + b(a + b - c) + c(a + b - c) \]
This expands to:
\[ a^2 + ab - ac + ab + b^2 - bc + ac + bc - c^2 \]
Combine like terms:
\[ a^2 + 2ab + b^2 - c^2 \]

**Final Answer**: \( a^2 + 2ab + b^2 - c^2 \)

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