1. Solve the linear equation $4x + 3 = 2x + 9$. Subtract $2x$ from both sides: $4x - 2x + 3 = 9$. Simplify: $2x + 3 = 9$. Subtract 3: $2x = 6$. Divide by 2: $x = 3$. 2. Solve the quadratic equation $5x^2 = 3x$. Rearrange: $5x^2 - 3x = 0$. Factor: $x(5x - 3) = 0$. Set each factor to zero: $x=0$ or $5x-3=0$. Solve $5x-3=0$: $x=\frac{3}{5}$. 3. Solve $50 - x^2 = 25 - x - x^2$. Add $x^2$ to both sides: $50 = 25 - x$. Subtract 25: $25 = -x$. Multiply by -1: $x = -25$. 4. Solve $x^2 - 11x - 12 = 0$. Factor: $(x - 12)(x + 1) = 0$. Solutions: $x=12$ or $x=-1$. 5. Solve $x^2 + 10x = 0$. Factor: $x(x + 10) = 0$. Solutions: $x=0$ or $x=-10$. 6. Solve $3(3x + 5) + 6 = 3$. Expand: $9x + 15 + 6 = 3$. Simplify: $9x + 21 = 3$. Subtract 21: $9x = -18$. Divide by 9: $x = -2$. 7. Solve $(y + 2)^2 = y^2 + 13$. Expand left: $y^2 + 4y + 4 = y^2 + 13$. Subtract $y^2$: $4y + 4 = 13$. Subtract 4: $4y = 9$. Divide by 4: $y = \frac{9}{4}$. 8. Solve $8x^2 - 2x - 3 = 0$ using quadratic formula. $a=8$, $b=-2$, $c=-3$. Discriminant: $\Delta = (-2)^2 - 4(8)(-3) = 4 + 96 = 100$. Roots: $x = \frac{2 \pm \sqrt{100}}{16} = \frac{2 \pm 10}{16}$. Solutions: $x=\frac{12}{16} = \frac{3}{4}$ or $x=\frac{-8}{16} = -\frac{1}{2}$. 9. Solve $3x^2 + 27x + 42 = 0$. Divide by 3: $x^2 + 9x + 14 = 0$. Factor: $(x + 7)(x + 2) = 0$. Solutions: $x=-7$ or $x=-2$. 10. Solve $2x - 9 = \frac{x}{4}$. Multiply both sides by 4: $8x - 36 = x$. Subtract $x$: $7x - 36 = 0$. Add 36: $7x = 36$. Divide by 7: $x = \frac{36}{7}$. 11. Solve $(a + 2)(a - 4) = (a + 3)^2$. Expand left: $a^2 - 4a + 2a - 8 = a^2 + 6a + 9$. Simplify left: $a^2 - 2a - 8 = a^2 + 6a + 9$. Subtract $a^2$: $-2a - 8 = 6a + 9$. Add $2a$: $-8 = 8a + 9$. Subtract 9: $-17 = 8a$. Divide by 8: $a = -\frac{17}{8}$. 12. Solve $6x^2 = x + 2$. Rearrange: $6x^2 - x - 2 = 0$. Use quadratic formula: $a=6$, $b=-1$, $c=-2$. Discriminant: $1 + 48 = 49$. Roots: $x = \frac{1 \pm 7}{12}$. Solutions: $x=\frac{8}{12} = \frac{2}{3}$ or $x=\frac{-6}{12} = -\frac{1}{2}$. 13. Solve $(x - 3)(x + 5) = -16$. Expand left: $x^2 + 5x - 3x - 15 = -16$. Simplify: $x^2 + 2x - 15 = -16$. Add 16: $x^2 + 2x + 1 = 0$. Factor: $(x + 1)^2 = 0$. Solution: $x = -1$. 14. Solve $(b + 5)^2 = (b + 1)^2$. Take square root: $b + 5 = \pm (b + 1)$. Case 1: $b + 5 = b + 1$ gives $5=1$ (no). Case 2: $b + 5 = -b - 1$. Add $b$: $2b + 5 = -1$. Subtract 5: $2b = -6$. Divide by 2: $b = -3$. 15. Solve $x^3 - 6x^2 + 8x = 0$. Factor: $x(x^2 - 6x + 8) = 0$. Solve $x=0$ or $x^2 - 6x + 8=0$. Factor quadratic: $(x - 4)(x - 2) = 0$. Solutions: $x=0$, $x=4$, $x=2$. 16. Solve $4x(x + 1) = 3$. Expand: $4x^2 + 4x = 3$. Rearrange: $4x^2 + 4x - 3 = 0$. Use quadratic formula: $a=4$, $b=4$, $c=-3$. Discriminant: $16 + 48 = 64$. Roots: $x = \frac{-4 \pm 8}{8}$. Solutions: $x=\frac{4}{8} = \frac{1}{2}$ or $x=\frac{-12}{8} = -\frac{3}{2}$. 17. Solve $(2y + 3)^2 = (y + 1)(y - 3) + 3y^2$. Expand left: $4y^2 + 12y + 9$. Expand right: $y^2 - 3y + 3y^2 = 4y^2 - 3y$. Equation: $4y^2 + 12y + 9 = 4y^2 - 3y$. Subtract $4y^2$: $12y + 9 = -3y$. Add $3y$: $15y + 9 = 0$. Subtract 9: $15y = -9$. Divide by 15: $y = -\frac{3}{5}$. 18. Solve $-2x^2 = -8x + 6$. Rearrange: $-2x^2 + 8x - 6 = 0$. Divide by -2: $x^2 - 4x + 3 = 0$. Factor: $(x - 3)(x - 1) = 0$. Solutions: $x=3$ or $x=1$. 19. Solve $(5m + 2)^2 - 1 = (m - 3)(m + 5) + 24m^2$. Expand left: $25m^2 + 20m + 4 - 1 = 25m^2 + 20m + 3$. Expand right: $m^2 + 2m - 15 + 24m^2 = 25m^2 + 2m - 15$. Equation: $25m^2 + 20m + 3 = 25m^2 + 2m - 15$. Subtract $25m^2$: $20m + 3 = 2m - 15$. Subtract $2m$: $18m + 3 = -15$. Subtract 3: $18m = -18$. Divide by 18: $m = -1$. Final answers: 1. $x=3$ 2. $x=0$, $x=\frac{3}{5}$ 3. $x=-25$ 4. $x=12$, $x=-1$ 5. $x=0$, $x=-10$ 6. $x=-2$ 7. $y=\frac{9}{4}$ 8. $x=\frac{3}{4}$, $x=-\frac{1}{2}$ 9. $x=-7$, $x=-2$ 10. $x=\frac{36}{7}$ 11. $a=-\frac{17}{8}$ 12. $x=\frac{2}{3}$, $x=-\frac{1}{2}$ 13. $x=-1$ 14. $b=-3$ 15. $x=0$, $x=4$, $x=2$ 16. $x=\frac{1}{2}$, $x=-\frac{3}{2}$ 17. $y=-\frac{3}{5}$ 18. $x=3$, $x=1$ 19. $m=-1$