8. 计算:
(1)$(x - 2y)^2 - x(x - 4y)$; (2)$x(x + 2) + (x + 1)^2 - 4x$;
(3)$(3x + y)^2 + (x + y)(x - y)$; (4)$(x + 2y)^2 + (x - 2y)(x + 2y) + x(x - 4y)$.
答案：解:
(1)原式$=x^{2}-4xy+4y^{2}-x^{2}+4xy=4y^{2}$.
(2)原式$=x^{2}+2x+x^{2}+2x+1-4x=2x^{2}+1$.
(3)原式$=9x^{2}+6xy+y^{2}+x^{2}-y^{2}=10x^{2}+6xy$.
(4)原式$=x^{2}+4xy+4y^{2}+x^{2}-4y^{2}+x^{2}-4xy=3x^{2}$.
9. 先化简,再求值:
(1)$(2x - 3y)(2x + 3y) - (2x - y)^2$,其中$x = 3,y = 1$;
(2)$5a(a - 2) - (2a - 3)(2a + 3) - (a + 1)^2$,其中$a = -\frac{1}{3}$.
答案：解:
(1)原式$=4x^{2}-9y^{2}-(4x^{2}-4xy+y^{2})=4x^{2}-9y^{2}-$
$4x^{2}+4xy-y^{2}=4xy-10y^{2}$,
将$x=3,y=1$代入得,原式$=4×3×1-10×1^{2}=12-10$
$=2$.
(2)原式$=5a^{2}-10a-(4a^{2}-9)-(a^{2}+2a+1)=5a^{2}-$
$10a-4a^{2}+9-a^{2}-2a-1=-12a+8$,
当$a=-\frac{1}{3}$时,原式$=-12×(-\frac{1}{3})+8=4+8=12$.
10. (2024 春·耒阳期末)已知$(x - y)^2 = 4,(x + y)^2 = 64$,求下列各代数式的值:
(1)$x^2 + y^2$; (2)$xy$.
答案：解:
(1)$(x-y)^{2}+(x+y)^{2}=x^{2}-2xy+y^{2}+x^{2}+2xy+$
$y^{2}=2x^{2}+2y^{2}=4+64=68$,$\therefore x^{2}+y^{2}=34$.
(2)$(x+y)^{2}-(x-y)^{2}=4xy=64-4=60$,$\therefore xy=15$.
11. (2024 春·南京期末)已知$a - c - b = -10,(a - b)c = -12$,求$(a - b)^2 + c^2$的值.
答案：解:$(a-b)^{2}+c^{2}$
$=[(a-b)-c]^{2}+2(a-b)c=(-10)^{2}+2×(-12)$
$=100-24=76$.
