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Please provide step-by-step solutions for  all the questions  in Task 2: Expanding and Factorising and provide the output in pdf file 

I will provide detailed step-by-step solutions for each question in Task 2: Expanding and Factorising, as found in the document. Here's the plan:

1. **Solve all the questions** in Task 2, breaking down each step clearly.
2. **Compile the solutions into a PDF** for easy reference.

The Task 2 section includes problems related to expanding and factorising expressions, which involve algebraic manipulation, such as expanding brackets, simplifying, and factorising polynomials.

### List of Questions in Task 2:
1. \(x^2 + ax + b \equiv (x - 5)^2 + 7\)
2. \(2x^2 - 2bx + 7a \equiv 2(x - a)^2 + 3\)
3. \(x^3 + ax^2 + bx + 150\) factorises to \((x + c)^2(x + d)\)
4. Expand and simplify \(xy(2x + 3y)(5x - 2y)\)
5. Expand and simplify \((x + 2)(x + 3)(x + 4)\)
6. Expand and simplify \((x - 5)(x^2 + 4x - 2)\)
7. Simplify \((3n + 1)^2 - (3n - 1)^2\)
8. \(3a(2x - 1) + 4(ax + 5) \equiv 60x + b\)
9. Write \(q\) and \(r\) in terms of \(m\) and \(n\) for \(x^2 + qx + r \equiv (x + m)(x + n)\)
10. Expand and simplify \(c^{3/2}(c^{1/2} + c^{-3/2})\)
11. \(5(3x - 2) - 3(x - h) \

Explore detailed step-by-step solutions for expanding and factorising problems in algebra. Includes exercises on expanding brackets, simplifying expressions, and factorising polynomials. Ideal for high school students studying algebra.

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