\(\left(3x+4\right)^2-10x-\left(x-4\right)\left(x+4\right)\)

\(=9x^2+24x+16-10x-x^2+16\)

\(=8x^2+14x\)

P = ( 3x + 4 )² - 10x - ( x - 4 )( x + 4 )

P = 9x² + 24x + 16 - 10x - ( x² - 16 )

P = 9x² + 24x + 16 - 10x - x² + 16

P = 8x² + 14x + 32

P = 2( 4x² + 7x + 16 )

a) Để rút gọn biểu thức (x+2)(x^2+4x+4)-(x-2)(x^2-4x-4)-12x^2-x, ta thực hiện các bước sau:

(x+2)(x^2+4x+4) = x(x^2+4x+4) + 2(x^2+4x+4)
= x^3 + 4x^2 + 4x + 2x^2 + 8x + 8
= x^3 + 6x^2 + 12x + 8

(x-2)(x^2-4x-4) = x(x^2-4x-4) - 2(x^2-4x-4)
= x^3 - 4x^2 - 4x - 2x^2 + 8x + 8
= x^3 - 6x^2 + 4x + 8

Thay vào biểu thức ban đầu, ta có:
(x+2)(x^2+4x+4)-(x-2)(x^2-4x-4)-12x^2-x
= (x^3 + 6x^2 + 12x + 8 - (x^3 - 6x^2 + 4x - 12x^2 - x
= x^3 + 6x^2 + 12x + 8 - x^3 + 6x^2 - 4x - 8 - 12x^2 - x
= 8x + 8 - 4x - 8
= 4x

Vậy biểu thức đã được rút gọn thành 4x.

b) Để rút gọn biểu thức (x-2)(x+2)(x+3)-(x+1)(x^2-x+1), ta thực hiện các bước sau:

(x-2)(x+2) = x^2 - 2^2 = x^2 - 4

Thay vào biểu thức ban đầu, ta có:
(x-2)(x+2)(x+3)-(x+1)(x^2-x+1)
= (x^2 - 4)(x+3) - (x+1)(x^2-x+1)
= x^3 + 3x^2 - 4x - 12 - (x^3 + x^2 - x + x^2 - x + 1)
= x^3 + 3x^2 - 4x - 12 - x^3 - x^2 + x - x^2 + x - 1
= x^3 - x^3 + 3x^2 - x^2 - x^2 + 3x - 4x + x - 12 - 1
= 2x^2 - x - 13

Vậy biểu thức đã được rút gọn thành 2x^2 - x - 13.

\(A=\frac{x^3-3x^2-7x-15}{x^5-x^4-10x^3-38x^2-51x-45}\)

\(=\frac{x^2\left(x-5\right)+2x\left(x-5\right)+3\left(x-5\right)}{x^4\left(x-5\right)+4x^3\left(x-5\right)+10x^2\left(x-5\right)+12x\left(x-5\right)+9\left(x-5\right)}\)

\(=\frac{\left(x-5\right)\left(x^2+2x+3\right)}{\left(x-5\right)\left(x^4+4x^3+10x^2+12x+9\right)}\)

\(=\frac{x^2+2x+3}{x^4+4x^3+10x^2+12x+9}\)

\(=\frac{x^2+2x+3}{\left(x^2\right)^2+2.x^2.2x+\left(2x\right)^2+6x^2+12x+9}\)

\(=\frac{x^2+2x+3}{\left(x^2+2x\right)^2+2.\left(x^2+2x\right).3+3^2}\)

\(=\frac{\left(x^2+2x+3\right)}{\left(x^2+2x+3\right)^2}=\frac{1}{x^2+2x+3}\)

b, \(A=\frac{1}{x^2+2x+3}=\frac{1}{\left(x+1\right)^2+2}\le\frac{1}{2}\forall x\)

Dấu "=" xảy ra khi: \(x+1=0\Rightarrow x=-1\)

Vậy GTLN của A là \(\frac{1}{2}\) khi x = -1

(x-3)(x² + 3x +9)- ((x-4)((x+4)=21
x³ - 27 - x² + 4 = 21
x² + x - 27 -x² + 4 =21
x=27 -4 + 21
x= 44

2)

( x _ 3 ) ( x^2 + 3x + 9 ) - ( x - 4 ) . ( x + 4 ) = 21

= x^3 - 9 - x^2 - 2^2 = 21

= x - 9 - x^2 - 4

=  x^3 - x^2  - 9 - 4

x = - 9 - 4 = - 13

Ý làm sai rùi : làm lại :

( x - 3 ) ( x^2 + 3x + 9 ) - ( x - 4 ) . ( x + 4 ) = 21

= x^3 - 27 - x^2 - 2^2 = 21

= x^3 - x^2 - 27 - 4 = 21

              x - 23 = 21

              x      = 21 + 23 = 22

P/s : ko chắc 

\(a,\left(a+2\right)^2-\left(a+2\right)\left(a-2\right)\)

\(=a^2+4x+4-a^2+4\)

\(=4x+8\)

\(=4\left(x+2\right)\)

\(b,\left(a+b\right)^2-\left(a-b\right)^2\)

\(=a^2+2ab+b^2-\left(a^2-2ab+b^2\right)\)

\(=a^2+2ab+b^2-a^2+2ab-b^2\)

\(=4ab\)

\(c,\left(3x+4\right)^2-10x-\left(x+4\right)\left(x-4\right)\)

\(=9x^2+24x+16-10x-x^2+16\)

\(=8x^2+14x+32\)

\(=2\left(4x^2+7x+16\right)\)

thanks ban nha ^^

a)(3x+4)²-10x-(x-4)(x+4)

    9x²+24x+16-10x-x²+16

    8x²+14x+32

b)(x+1)(x-2)(x²+1)(x+2)(x-1)(x²+4)

   (x+1)(x-1)(x+2)(x-2)(x²+1)(x²+4)

    (x²-1)(x²-4)(7x²+4)

    (-3x²+4)(7x²+4)

    -21x²-12x²+28x²+16

    16-x²

a)(3x+4)2-10x-(x-4)(x+4)

9x2+24x+16-10x-x2+16

8x2+14x+32

b)(x+1)(x-2)(x2+1)(x+2)(x-1)(x2+4)

(x+1)(x-1)(x+2)(x-2)(x2+1)(x2+4)

(x2-1)(x2-4)(7x2+4)

(-3x2+4)(7x2+4)

-21x2-12x2+28x2+16

16-x2

B1

A=11x^2-x-2

B=2(-4+x)

B2

a)=(x+3)^2(x-3)

B2

b)=(5x-7)(x+2)

Ta có: \(\frac{\left(x^2\right)^2-10x^2+9}{x^4+6x^3+9x^2+2x^3+12x^2+18x+x^2+6x+9}\)

=  \(\frac{\left(x^2-1\right)\left(x^2-3\right)}{x^2\left(x^2+6x+9\right)+2x\left(x^2+6x+9\right)+\left(x^2+6x+9\right)}\)

=  \(\frac{\left(x-1\right)\left(x+1\right)\left(x-3\right)\left(x+3\right)}{\left(x^2+6x+9\right)\left(x^2+2x+1\right)}\)

=  \(\frac{\left(x-1\right)\left(x+1\right)\left(x-3\right)\left(x+3\right)}{\left(x+3\right)^2.\left(x+1\right)^2}\)

=  \(\frac{\left(x-1\right)\left(x+1\right)\left(x-3\right)\left(x+3\right)}{\left(x+3\right)\left(x+3\right)\left(x+1\right)\left(x+1\right)}\)

=  \(\frac{\left(x-1\right)\left(x-3\right)}{\left(x+1\right)\left(x+3\right)}\)

D = (3x - 2)^2 - 3(x - 4)(4 + x) + (x - 3)^3 - (x^2 - x + 1)(x + 1)

D = 9x^2 - 12x + 4 - 3x^2 + 48 + x^3 - 9x^2 + 27x - 27 - x^3 - 1

D = -3x^2 + 15x + 24