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Let's solve the questions one by one:

**7. The factorisation of \(4x^2 + 8x + 3\) is:**

We need to factor the quadratic expression \(4x^2 + 8x + 3\).

Using the method of splitting the middle term or applying the quadratic formula:

- First, check if we can find two numbers that multiply to \(4 \times 3 = 12\) and add up to \(8\). These numbers are \(6\) and \(2\).

- So, we can rewrite the middle term as:
  \[
  4x^2 + 6x + 2x + 3
  \]

- Now, factor by grouping:
  \[
  2x(2x + 3) + 1(2x + 3)
  \]

- This simplifies to:
  \[
  (2x + 1)(2x + 3)
  \]

Thus, the correct factorization is \((2x + 1)(2x + 3)\), which corresponds to option **(b)**.

---

**8. Which of the following is a factor of \((x + y)^3 - (x^3 + y^3)\)?**

We start by simplifying \((x + y)^3 - (x^3 + y^3)\):

- The expression \((x + y)^3\) expands to:
  \[
  x^3 + 3x^2y + 3xy^2 + y^3
  \]

- Subtract \((x^3 + y^3)\) from this:
  \[
  (x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 + y^3) = 3x^2y + 3xy^2 = 3xy(x + y)
  \]

So, the factor is \(3xy(x + y)\), and the correct answer is **(d)**.

---

**9. Euclid stated that all right angles are equal to each other in the form of:**

This statement is one of Euclid's postulates from his work *Elements*. Specifically, Euclid's fourth postulate states that all right angles are equal to one another.

Thus, the correct answer is **(c)**, a postulate.

---

### To summarize:
- **7**: Answer is **(b)** \((2x+1)(2x+3)\)
- **8**: Answer is **(d)** \(3xy\)
- **9**: Answer is **(c)** a postulate

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Would you like a more detailed explanation on any of these? Here are five related questions for practice:

1. How would you factor \(6x^2 + 13x + 6\)?
2. What is the factorization of \((x - 1)^3 - (x^3 - 1)\)?
3. Can all quadratic equations be factored without using the quadratic formula?
4. What is another example of a postulate from Euclid's *Elements*?
5. How can you expand \((x + y)^4\)?

**Tip**: When factoring quadratics, always check if you can split the middle term for easier factorization.

Explore how to factor the quadratic equation 4x^2 + 8x + 3, understand factorization of (x + y)^3 - (x^3 + y^3), and learn about Euclid's fourth postulate that all right angles are equal. Suitable for students in Grades 8-10.

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