a) $3x(2-x)+2x(x-1)=5x(x+3)$
$6x - 3x^2 + 2x^2 - 2x = 5x^2 + 15x$
$-x^2 + 4x = 5x^2 + 15x$
$0 = 5x^2 + x^2 + 15x - 4x$
$0 = 6x^2 + 11x$
$x(6x+11) = 0$
$\Leftrightarrow x = 0$ hoặc $6x+11 = 0$
$\Leftrightarrow x = 0$ hoặc $6x = -11$
$\Leftrightarrow x = 0$ hoặc $x = -\frac{11}{6}$
Vậy $S = \{0; -\frac{11}{6}\}$.

b) $3x(x+1)-5x(3-x)+6(x^2+2x+3)=0$
$3x^2 + 3x - 15x + 5x^2 + 6x^2 + 12x + 18 = 0$
$(3x^2 + 5x^2 + 6x^2) + (3x - 15x + 12x) + 18 = 0$
$14x^2 + 0x + 18 = 0$
$14x^2 + 18 = 0$
$14x^2 = -18$
$x^2 = -\frac{18}{14} = -\frac{9}{7}$
Vì $x^2 \ge 0$ với mọi $x$, nên $x^2 = -\frac{9}{7}$ là vô nghiệm.
Vậy phương trình vô nghiệm.

c) $2(x+1)^2-(x-3)(x+3)-(x-4)^2=0$
$2(x^2+2x+1) - (x^2-9) - (x^2-8x+16) = 0$
$2x^2 + 4x + 2 - x^2 + 9 - x^2 + 8x - 16 = 0$
$(2x^2 - x^2 - x^2) + (4x + 8x) + (2 + 9 - 16) = 0$
$0x^2 + 12x - 5 = 0$
$12x - 5 = 0$
$12x = 5$
$x = \frac{5}{12}$
Vậy $S = \{\frac{5}{12}\}$.

d) $(x-5)^2-x(x-4)=9$
$x^2 - 10x + 25 - x^2 + 4x = 9$
$(x^2 - x^2) + (-10x + 4x) + 25 = 9$
$-6x + 25 = 9$
$-6x = 9 - 25$
$-6x = -16$
$x = \frac{-16}{-6} = \frac{8}{3}$
Vậy $S = \{\frac{8}{3}\}$.

e) $(x-5)^2+(x-4)(1-x)=0$
$x^2 - 10x + 25 + x - x^2 - 4 + 4x = 0$
$(x^2 - x^2) + (-10x + x + 4x) + (25 - 4) = 0$
$-5x + 21 = 0$
$-5x = -21$
$x = \frac{21}{5}$
Vậy $S = \{\frac{21}{5}\}$.

e) $(2x+1)^2-4(x+2)^2=9$
$(2x+1)^2 - (2(x+2))^2 = 9$
$(2x+1)^2 - (2x+4)^2 = 9$
Áp dụng hđt $A^2 - B^2 = (A-B)(A+B)$:
$((2x+1) - (2x+4))((2x+1) + (2x+4)) = 9$
$(2x+1 - 2x - 4)(2x+1+2x+4) = 9$
$(-3)(4x+5) = 9$
$4x+5 = \frac{9}{-3}$
$4x+5 = -3$
$4x = -3 - 5$
$4x = -8$
$x = -2$
Vậy $S = \{-2\}$.

`f)` $(x+3)^2-(x-4)(x+8)=1$
$x^2 + 6x + 9 - (x^2 + 8x - 4x - 32) = 1$
$x^2 + 6x + 9 - (x^2 + 4x - 32) = 1$
$x^2 + 6x + 9 - x^2 - 4x + 32 = 1$
$(x^2 - x^2) + (6x - 4x) + (9 + 32) = 1$
$2x + 41 = 1$
$2x = 1 - 41$
$2x = -40$
$x = -20$
Vậy $S = \{-20\}$.

`g)` $3(x+2)^2+(2x-1)^2-7(x+3)(x-3)=36$
$3(x^2+4x+4) + (4x^2-4x+1) - 7(x^2-9) = 36$
$3x^2 + 12x + 12 + 4x^2 - 4x + 1 - 7x^2 + 63 = 36$
$(3x^2 + 4x^2 - 7x^2) + (12x - 4x) + (12 + 1 + 63) = 36$
$0x^2 + 8x + 76 = 36$
$8x + 76 = 36$
$8x = 36 - 76$
$8x = -40$
$x = -5$
Vậy $S = \{-5\}$.