1. **Problem a:** Write $-x^2 - 16x + 7$ in the form $-(x + c)^2 + d$, where $c$ and $d$ are integers. 2. **Step 1:** Start with the expression: $$-x^2 - 16x + 7$$ 3. **Step 2:** Factor out the negative sign from the $x^2$ and $x$ terms: $$-(x^2 + 16x) + 7$$ 4. **Step 3:** Complete the square inside the parentheses. Take half of 16, which is 8, and square it to get 64. 5. **Step 4:** Add and subtract 64 inside the parentheses: $$-(x^2 + 16x + 64 - 64) + 7 = -( (x + 8)^2 - 64 ) + 7$$ 6. **Step 5:** Distribute the negative sign: $$-(x + 8)^2 + 64 + 7 = -(x + 8)^2 + 71$$ 7. **Answer a:** So, $c = 8$ and $d = 71$. --- 8. **Problem b:** Write down the coordinates of the turning point of the curve $y = -x^2 - 16x + 7$. 9. **Step 1:** The vertex form is $y = -(x + 8)^2 + 71$. 10. **Step 2:** The turning point (vertex) is at $(-c, d) = (-8, 71)$. --- 11. **Problem c:** Express $2x^2 + 16x + 11$ in the form $a(x + b)^2 + c$, where $a$, $b$, and $c$ are integers. 12. **Step 1:** Factor out 2 from the $x^2$ and $x$ terms: $$2(x^2 + 8x) + 11$$ 13. **Step 2:** Complete the square inside the parentheses. Half of 8 is 4, squared is 16. 14. **Step 3:** Add and subtract 16 inside the parentheses: $$2(x^2 + 8x + 16 - 16) + 11 = 2((x + 4)^2 - 16) + 11$$ 15. **Step 4:** Distribute 2: $$2(x + 4)^2 - 32 + 11 = 2(x + 4)^2 - 21$$ 16. **Answer c:** $a = 2$, $b = 4$, $c = -21$. --- 17. **Problem d:** Write $-2x^2 - 16x - 10$ in the form $a(x + b)^2 + c$, where $a$, $b$, and $c$ are integers. 18. **Step 1:** Factor out $-2$ from the $x^2$ and $x$ terms: $$-2(x^2 + 8x) - 10$$ 19. **Step 2:** Complete the square inside the parentheses. Half of 8 is 4, squared is 16. 20. **Step 3:** Add and subtract 16 inside the parentheses: $$-2(x^2 + 8x + 16 - 16) - 10 = -2((x + 4)^2 - 16) - 10$$ 21. **Step 4:** Distribute $-2$: $$-2(x + 4)^2 + 32 - 10 = -2(x + 4)^2 + 22$$ 22. **Answer d:** $a = -2$, $b = 4$, $c = 22$. --- 23. **Problem e:** Write down the coordinates of the turning point of the curve $y = -2x^2 - 16x - 10$. 24. **Step 1:** The vertex form is $y = -2(x + 4)^2 + 22$. 25. **Step 2:** The turning point is at $(-b, c) = (-4, 22)$. --- 26. **Problem f:** Find the perimeter of the piece of card with a photo 18 cm by 15 cm and a 3 cm border. 27. **Step 1:** Total length = $18 + 3 + 3 = 24$ cm. 28. **Step 2:** Total width = $15 + 3 + 3 = 21$ cm. 29. **Step 3:** Perimeter = $2 imes (24 + 21) = 2 imes 45 = 90$ cm. --- 30. **Problem g:** Probability that both biscuits chosen are the same type from 9 bourbon and 2 custard cream biscuits. 31. **Step 1:** Total biscuits = 11. 32. **Step 2:** Probability first biscuit is bourbon = $\frac{9}{11}$. 33. **Step 3:** Probability second biscuit is bourbon given first was bourbon = $\frac{8}{10}$. 34. **Step 4:** Probability both bourbon = $\frac{9}{11} \times \frac{8}{10} = \frac{72}{110} = \frac{36}{55}$. 35. **Step 5:** Probability first biscuit is custard cream = $\frac{2}{11}$. 36. **Step 6:** Probability second biscuit is custard cream given first was custard cream = $\frac{1}{10}$. 37. **Step 7:** Probability both custard cream = $\frac{2}{11} \times \frac{1}{10} = \frac{2}{110} = \frac{1}{55}$. 38. **Step 8:** Probability both biscuits are the same type = $\frac{36}{55} + \frac{1}{55} = \frac{37}{55}$.