Factorise the following quadratic expressions: a) 8x2 + 6x + 1 b) 3x2 - 8x + 5 c) 6x2 + 11x - 10

You've done a great job with these factorisation problems! Let's go through them.

a) Factorise $8x^2 + 6x + 1$

Your work:
Sum ($S$) = 6
Product ($P$) = $8 \times 1 = 8$
The two numbers that multiply to 8 and add to 6 are 4 and 2.

$$8x^2 + 4x + 2x + 1$$ $$(8x^2 + 4x) + (2x + 1)$$ $$4x(2x + 1) + 1(2x + 1)$$ $$(2x + 1)(4x + 1)$$

✅ Correct.
The factorised expression is $\boxed{(2x + 1)(4x + 1)}$.

b) Factorise $3x^2 - 8x + 5$

Your work:
Sum ($S$) = -8
Product ($P$) = $3 \times 5 = 15$
The two numbers that multiply to 15 and add to -8 are -3 and -5.

$$3x^2 - 3x - 5x + 5$$ $$(3x^2 - 3x) + (-5x + 5)$$ $$3x(x - 1) - 5(x - 1)$$ $$(x - 1)(3x - 5)$$

✅ Correct. Your final answer is correct. Just a minor note on the grouping step you wrote: $(3x^2 - 3x) - (5x + 5)$ would expand to $3x^2 - 3x - 5x - 5$, which is not the original expression. It should be $(3x^2 - 3x) + (-5x + 5)$ or $3x^2 - 3x - 5x + 5$ directly. However, you correctly factored $3x(x-1) - 5(x-1)$, so the final result is correct.
The factorised expression is $\boxed{(x - 1)(3x - 5)}$.

c) Factorise $6x^2 + 11x - 10$

Your work:
Sum ($S$) = 11
Product ($P$) = $6 \times (-10) = -60$
The two numbers that multiply to -60 and add to 11 are 15 and -4.

$$6x^2 + 15x - 4x - 10$$ $$(6x^2 - 4x) + (15x - 10)$$ $$2x(3x - 2) + 5(3x - 2)$$ $$(3x - 2)(2x + 5)$$

✅ Correct.
The factorised expression is $\boxed{(3x - 2)(2x + 5)}$.

d) Factorise $3x^2 + 7x - 6$

Your work:
Sum ($S$) = 7
Product ($P$) = $3 \times (-6) = -18$
The two numbers that multiply to -18 and add to 7 are 9 and -2.

$$3x^2 + 9x - 2x - 6$$ $$(3x^2 - 2x) + (9x - 6)$$ $$x(3x - 2) + 3(3x - 2)$$ $$(x + 3)(3x - 2)$$

✅ Correct.
The factorised expression is $\boxed{(x + 3)(3x - 2)}$.

e) Factorise $2x^2 + 5x + 3$

Your work:
Sum ($S$) = 5
Product ($P$) = $2 \times 3 = 6$
The two numbers that multiply to 6 and add to 5 are 2 and 3.

$$2x^2 + 2x + 3x + 3$$ $$(2x^2 + 2x) + (3x + 3)$$ $$2x(x + 1) + 3(x + 1)$$ $$(2x + 3)(x + 1)$$

✅ Correct.
The factorised expression is $\boxed{(2x + 3)(x + 1)}$.