Simplify.
(i) (x²-5) (x+5)+25
(ii) (a²+5) (b³+3)+5
(iii) (t+s²) (t²-s)
(iv) (a+b)(c-d)+(9a-b)(c+d)+2(ac+bd)
(v) (x+y)(2x+y)+(x+2y)+(x+2y)(x-y)`

Explanation

Let's simplify the given expressions step by step.

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

1. **Expand the expression**:
\[ (x^2 - 5)(x + 5) = x^2 \cdot x + x^2 \cdot 5 - 5 \cdot x - 5 \cdot 5 \]
This simplifies to:
\[ x^3 + 5x^2 - 5x - 25 \]

2. **Add 25**:
\[ x^3 + 5x^2 - 5x - 25 + 25 \]
The \(-25\) and \(+25\) cancel each other out:
\[ x^3 + 5x^2 - 5x \]

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

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

1. **Expand the expression**:
\[ (a^2 + 5)(b^3 + 3) = a^2 \cdot b^3 + a^2 \cdot 3 + 5 \cdot b^3 + 5 \cdot 3 \]
This simplifies to:
\[ a^2b^3 + 3a^2 + 5b^3 + 15 \]

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

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

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

1. **Expand the expression**:
\[ (t + s^2)(t^2 - s) = t \cdot t^2 + t \cdot (-s) + s^2 \cdot t^2 + s^2 \cdot (-s) \]
This simplifies to:
\[ t^3 - ts + s^2t^2 - s^3 \]

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

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

1. **Expand 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 terms**:
\[ (ac - ad + bc - bd) + (9ac + 9ad - bc - bd) + (2ac + 2bd) \]
Combine like terms:
\[ (ac + 9ac + 2ac) + (-ad + 9ad) + (bc - bc) + (-bd - bd + 2bd) \]
This simplifies to:
\[ 12ac + 8ad - bd \]

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

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

1. **Expand the first term**:
\[ (x + y)(2x + y) = 2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2 \]

2. **Expand the second term**:
\[ (x + 2y)(x - y) = x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2 \]

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

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

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