1. **Decide if each statement is an equation, a formula, or an identity:** 1.a. $a^2 + b^2 = c^2$ is an **identity** (Pythagorean theorem, true for right triangles). 1.b. $2x + 5 = 15$ is an **equation** (solvable for $x$). 1.c. $2(4x - 7) = 8x - 14$ is an **identity** (true for all $x$ after expansion). 2. **Are these identities true or false?** 2.a. $6(p + 6) = 6p + 6$? Expand left: $6p + 36$; right: $6p + 6$; Not equal, so **false**. 2.b. $9(d - 2) = 9d - 18$? Expand left: $9d - 18$; right: $9d - 18$; Equal, so **true**. 2.c. $5t(t + 8) = 5t^2 + 40t$? Expand left: $5t^2 + 40t$; right: $5t^2 + 40t$; Equal, so **true**. 2.d. $2v(4v - 6) = 4v^2 - 12v$? Expand left: $8v^2 - 12v$; right: $4v^2 - 12v$; Not equal, so **false**. 3. **Expand and simplify:** 3.a. $4(m + 3) + 6(m - 4)$ = $4m + 12 + 6m - 24 = 10m - 12$ 3.b. $5(x - 7) + 4(x + 8)$ = $5x - 35 + 4x + 32 = 9x - 3$ 3.c. $v(v - 5) + 8(v + 5)$ = $v^2 - 5v + 8v + 40 = v^2 + 3v + 40$ 3.d. $3x(x + 6) - x(x + 4)$ = $3x^2 + 18x - x^2 - 4x = 2x^2 + 14x$ 4. **Expand, simplify, and factorise if possible:** 4.a. $(x + 5)^2 - 25$ = $(x^2 + 10x + 25) - 25 = x^2 + 10x = x(x + 10)$ 4.b. $(x - 3)^2 - 9$ = $(x^2 - 6x + 9) - 9 = x^2 - 6x = x(x - 6)$ 4.c. $(x - 4)^2 + 8x$ = $x^2 - 8x + 16 + 8x = x^2 + 16$ 4.d. $(x + 3)^2 - 2(x + 1)$ = $x^2 + 6x + 9 - 2x - 2 = x^2 + 4x + 7$ 4.e. $(x - 2)^2 + x(x - 1)$ = $x^2 - 4x + 4 + x^2 - x = 2x^2 - 5x + 4$ 4.f. $x(x^2 + 6) - (2x + 1)$ = $x^3 + 6x - 2x - 1 = x^3 + 4x - 1$ 5. **Fluency Expand:** 5.a. $x(x^2 + 6) = x^3 + 6x$ 5.b. $x(x^2 + 3x - 4) = x^3 + 3x^2 - 4x$ 5.c. $x(x^2 - 2x + 4) = x^3 - 2x^2 + 4x$ 5.d. $x(x - 2)(x + 3)$ First expand $(x - 2)(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6$ Then multiply by $x$: $x^3 + x^2 - 6x$ 5.e. $x(x + 2)(x - 4)$ Expand $(x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8$ Multiply by $x$: $x^3 - 2x^2 - 8x$ 5.f. $x(x - 5)(x - 3)$ Expand $(x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15$ Multiply by $x$: $x^3 - 8x^2 + 15x$ 6. **Problem solving: Complete** Given: $x(x + oxed{2})(x - 4) = x^3 - 2x^2 - oxed{8}x$ Check expansion: $(x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8$ Multiply by $x$: $x^3 - 2x^2 - 8x$ 7. **Expand and simplify:** 7.a. $(x + rac{1}{2})^2 = x^2 + 2 imes x imes rac{1}{2} + rac{1}{4} = x^2 + x + rac{1}{4}$ 7.b. $(x - rac{1}{4})^2 = x^2 - 2 imes x imes rac{1}{4} + rac{1}{16} = x^2 - rac{1}{2}x + rac{1}{16}$ 7.c. $(x + rac{1}{3})(x - 2) = x^2 - 2x + rac{1}{3}x - rac{2}{3} = x^2 - rac{5}{3}x - rac{2}{3}$ 8. **Show that:** $(x + rac{1}{4})(x + 3) = rac{1}{4}(4x^2 + 13x + 3)$ Expand left: $x^2 + 3x + rac{1}{4}x + rac{3}{4} = x^2 + rac{13}{4}x + rac{3}{4}$ Multiply right side: $rac{1}{4}(4x^2 + 13x + 3) = x^2 + rac{13}{4}x + rac{3}{4}$ Both sides equal, so identity holds. **Final answers provided for all parts as requested.**