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).

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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).

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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.

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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.