1. **State the problem:** Factorise the quadratic expression $$x^2 + 10x + 21$$. 2. **Recall the formula and rules:** To factorise a quadratic of the form $$x^2 + bx + c$$, we look for two numbers that multiply to $$c$$ and add to $$b$$. 3. **Identify values:** Here, $$b = 10$$ and $$c = 21$$. 4. **Find two numbers:** We need two numbers whose product is $$21$$ and sum is $$10$$. These numbers are $$3$$ and $$7$$ because $$3 \times 7 = 21$$ and $$3 + 7 = 10$$. 5. **Write the factorised form:** Using these numbers, the factorised form is: $$ (x + 3)(x + 7) $$ 6. **Verify by expansion:** Expanding: $$ (x + 3)(x + 7) = x^2 + 7x + 3x + 21 = x^2 + 10x + 21 $$ This matches the original quadratic, confirming the factorisation is correct. **Final answer:** $$ (x + 3)(x + 7) $$