Ta có : \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)

\(=\left(\left(x+1\right)\left(x+4\right)\right)\left(\left(x+2\right)\left(x+3\right)\right)-24\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\)

- Đặt \(x^2+5x+5=a\)

\(=\left(a-1\right)\left(a+1\right)-24=a^2-1-24=a^2-25\)

\(=\left(a-5\right)\left(a+5\right)\)

Ta có: \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\)

\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)+24-24\)

\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)\)

\(=\left(x^2+5x\right)\left(x^2+5x+10\right)\)

\(=x\left(x+5\right)\left(x^2+5x+10\right)\)

Bài làm:

a) \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+1\)

\(=\left[\left(x+1\right)\left(x+4\right)\right]\left[\left(x+2\right)\left(x+3\right)\right]+1\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)+1\)

Đặt \(x^2+5x+5=t\)\(\Rightarrow\left(t-1\right)\left(t+1\right)+1=t^2-1+1=t^2\)

\(=\left(x^2+5x+5\right)^2\)

b) Tương tự như a phân tích và đặt ra được: \(t^2-1-24=t^2-25=\left(t-5\right)\left(t+5\right)\)

\(=\left(x^2+5x\right)\left(x^2+5x+10\right)=x\left(x+5\right)\left(x^2+5x+10\right)\)

c) \(\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)+15\)

\(=\left[\left(x+1\right)\left(x+7\right)\right]\left[\left(x+3\right)\left(x+5\right)\right]+15\)

\(=\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\)

Đặt \(x^2+8x+11=t\)\(\Rightarrow\left(t-4\right)\left(t+4\right)+15=t^2-16+15=t^2-1\)

\(=\left(t-1\right)\left(t+1\right)=\left(x^2+8x+10\right)\left(x^2+8x+12\right)\)

\(=\left(x^2+8x+10\right)\left(x+2\right)\left(x+6\right)\)

d) \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)

\(=\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)

\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)

Đặt \(x^2+7x+11=t\)\(\Rightarrow\left(t-1\right)\left(t+1\right)-24=t^2-1-24=t^2-25\)

\(=\left(t-5\right)\left(t+5\right)=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)

\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)

Làm mẫu cho 1 vd:

a, (x+1)(x+2)(x+3)(x+4)+1

\(=\left(x+1\right)\left(x+4\right)\left(x+2\right)\left(x+3\right)+1\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)+1\)(1)

Đặt \(y=x^2+5x+5\)

Khi đó ::

(1) = \(\left(y-1\right)\left(y+1\right)+1\)

\(=y^2-1+1=y^2\)

Thay vào ta được: \(\left(x^2+5x+5\right)^2\)

a) (x+1)(x+2)(x+3)(x+4)+1=[(x+1)(x+4)].[(x+2)(x+3)]+1=(x²+5x+4)(x²+5x+6)+1

đặt t=x²+5x+5 ta có đa thức (t-1)(t+1)+1=t²-1+1=t². mà t=x²+5x+5

=> (x+1)(x+2)(x+3)(x+4)+1=(x²+5x+5)²

b) (x+1)(x+2)(x+3)(x+4)-24. theo kết quả câu (a) ta được (x+1)(x+2)(x+3)(x+4)=(x²+5x+4)(x²+5x+6)

đặt t=x²+5x+5 ta có đa thức (t-1)(t+1)-24=t²-1-24=t²-25=(t-5)(t+5)

mà t=x²+5x+5 => (t-5)(t+5)=(x²+5x)(x²+5x+10)

c) (x+1)(x+3)(x+5)(x+7)+15=[(x+1)(x+7)].[(x+3)(x+5)]+15=(x²+8x+7)(x²+8x+15)+15

đặt x²+8x+11=t ta có đa thức (t-4)(t+4)+15=t²-16+15=t²-1=(t-1)(t+1)

mà t=x²+8x+11 => (t-1)(t+1)=(x²+8x-10)(x²+8x+12)

d) (x+2)(x+3)(x+4)(x+5)-24=[(x+2)(x+5)][(x+3)(x+4)]-24=(x²+7x+12)(x²+7x+10)-24

đặt t=x²+7x+11 ta có đa thức (t-1)(t+1)-24=t²-1-24=t²-25=(t+5)(t-5)

mà t=x²+7x+11 => (t-5)(t+5)=(x²+7x+6)(x²+7x+16)

= (x+1)(x+4)(x+2)(x+3)-24

= (x² +5x+4) (x² +5x+6)-24

  Đặt x² +5x+4 =a

=>(x² +5x+4)(x²+5x+6)-24

= a(a+2)-24 = a² +2a-24

= a² +6a-4a-24

= a(a+6) - 4(a+6) = (a-4)(a+6)

= (x² +5x+a-4)(x² +5x+4+6) = (x² +5x)(x² +5x+10)

Nhớ mình nha mình âm diểm rồi:

M=(x+2)(x+3)(x+4)(x+5)-24

M=(x²+3x+2x+6)(x²+5x+4x+20)-24

M=(x²+5x+6)(x²+9x+20)-24

M=x⁴+9x³+20x²+5x³ +14x+100x+6x²+54x+120-24

M=x⁴+14x³+26x²+168x+96

=(x+1)(x+4)(x+2)(x+3) - 24

=(x^2+5x+4)(x^2+5x+6) - 24

=(x^2+5x+5-1)(x^2+5x+5+1) - 24 [hằng đẳng thức a^2-b^2 nha] 

=(x^2+5x+5)^2-1^2-24

=(x^2+5x+5)^2 - 25

=(x^2+5x+5)^2 - 5^2

=(x^2+5x+5-5)(x^2+5x+5+5)

=(x^2+5x)(x^2+5x+10

\(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)

\(=\left(x+1\right)\left(x+4\right)\left(x+2\right)\left(x+3\right)\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\)

Đặt \(x^2+5x+4=t\)

\(\Rightarrow\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24=t\left(t+2\right)-24\)

\(=t^2+2t+1-25\)

\(=\left(t+1\right)^2-5^2\)

\(=\left(t+1+5\right)\left(t+1-5\right)\)

\(=\left(t+6\right)\left(t-4\right)\)

\(=x\left(x+5\right)\left(x^2+5x+10\right)\)

Ta có : \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\)\(\left(x+4\right)-24\)

= \(\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\) (*)

. Đặt \(x^2+5x+4=t\) (1)

(*) <=> \(t\left(t+2\right)-24=t^2+2t-24=\left(t-4\right)\left(t+6\right)\) (2)

Thay (1) vào (2) ta suy ra : \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\) \(\left(x+4\right)-24=\)\(\left(x^2+5x+4-4\right)\left(x^2+5x+4+6\right)\) = \(\left(x^2+5x\right)\left(x^2+5x+10\right)\) = \(x\left(x+5\right)\left(x^2+5x+10\right)\)

\(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)

\(=\left[\left(x+1\right)\left(x+4\right)\right].\left[\left(x+2\right)\left(x+3\right)\right]-24\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\)

\(=\left(x^2+5x+4\right)^2+2.\left(x^2+5x+4\right)+1-25\)

\(=\left(x^2+5x+4+1\right)^2-5^2\)

\(=\left(x^2+5x+5-5\right)\left(x^2+5x+5+5\right)\)

\(=\left(x^2+5x\right)\left(x^2+5x+10\right)\)

\(=x\left(x+5\right)\left(x^2+5x+10\right)\)

Ta có 

<=>(x+1)(x+4)(x+2)(x+3)-24

<=>(X^2+5x4)(x^2+5x+6)-24

Đặt x^2+5x+5=x    (1)

Ta có 

<=>(x+1)(x-1)-24

<=>x^2-25

Thay 1 vào x  ta có

(x^2+5x+5)^2-5^2

<=>(x^2+10)(x^2+5x)(dpcm)

\(1,\\ 1,=15\left(x+y\right)\\ 2,=4\left(2x-3y\right)\\ 3,=x\left(y-1\right)\\ 4,=2x\left(2x-3\right)\\ 2,\\ 1,=\left(x+y\right)\left(2-5a\right)\\ 2,=\left(x-5\right)\left(a^2-3\right)\\ 3,=\left(a-b\right)\left(4x+6xy\right)=2x\left(2+3y\right)\left(a-b\right)\\ 4,=\left(x-1\right)\left(3x+5\right)\\ 3,\\ A=13\left(87+12+1\right)=13\cdot100=1300\\ B=\left(x-3\right)\left(2x+y\right)=\left(13-3\right)\left(26+4\right)=10\cdot30=300\\ 4,\\ 1,\Rightarrow\left(x-5\right)\left(x-2\right)=0\Rightarrow\left[{}\begin{matrix}x=2\\x=5\end{matrix}\right.\\ 2,\Rightarrow\left(x-7\right)\left(x+2\right)=0\Rightarrow\left[{}\begin{matrix}x=7\\x=-2\end{matrix}\right.\\ 3,\Rightarrow\left(3x-1\right)\left(x-4\right)=0\Rightarrow\left[{}\begin{matrix}x=\dfrac{1}{3}\\x=4\end{matrix}\right.\\ 4,\Rightarrow\left(2x+3\right)\left(2x-1\right)=0\\ \Rightarrow\left[{}\begin{matrix}x=-\dfrac{3}{2}\\x=\dfrac{1}{2}\end{matrix}\right.\)

1/(x+2)(x+3)(x+4)(x+5)-24

=(x+2)(x+5)(x+3)(x+4)

=(x+2)(x-2+7)(x+3)(x-3+7)

=[(x+2)(x-2)+7x+14][(x+3)(x-3)+7x+21]

=(x²-4+7x+14)(x²-9+7x+21)

=(x²+10+7x)(x²+12+7x)

2/(x²+x)²+4(x²+x)-12

=(x²+x)²+4(x²+x)+2²-16

=(x²+x+2)²-4²

=(x²+x+2+4)(x²+x+2-4)

=(x²+x+6)(x²+x-2)

3/(x²+x+1)(x²+x+2)-12

=(x²+x+1)(x²+x+-1+3)-12

=(x²+x+1)(x²+x+-1)+3(x²+x+1)-12

=(x²+x)-1+3(x²+x)+3-12

=(x²+x)(x²+x+3)-10

làm đến đây thì mk bí, bạn giúp suy nghĩ nốt nha

4/nó là nhân tử sẵn rồi mà

\(3/\)

\(\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)

\(=\left(x^2+x+1\right)\left(x^2+x+1+1\right)-12\)

\(=\left(x^2+x+1\right)^2+x^2+x+1-12\)

\(=\left(x^2+x+1\right)^2+4\left(x^2+x+1\right)-3\left(x^2+x+1\right)-12\)

\(=\left(x^2+x+1\right)\left(x^2+x+1+4\right)-3\left(x^2+x+1+4\right)\)

\(=\left(x^2+x+1-3\right)\left(x^2+x+1+4\right)\)

\(=\left(x^2+x-2\right)\left(x^2+x+5\right)\)