(x+5)³-x³-125

=x³+5³-x³-5³

=0

1. \(\left(x+5\right)^3-x^3-125\)

\(=x^3+15x^2+75x+125-x^3-125\)

\(=15x^2+75x\)

2. \(x^3+6x^2+12x+8=0\)

\(\Leftrightarrow x^3+2x^2+4x^2+8x+4x+8=0\)

\(\Leftrightarrow x^2\left(x+2\right)+4x\left(x+2\right)+4\left(x+2\right)=0\)

\(\Leftrightarrow\left(x^2+4x+4\right)\left(x+2\right)=0\)

\(\Leftrightarrow\left(x+2\right)^2\left(x+2\right)=0\)

\(\Leftrightarrow\left(x+2\right)^3=0\)

\(\Leftrightarrow x+2=0\Leftrightarrow x=-2\)

5x² - 5xy + 4y - 4x

= 5x ( x - y ) - 4 ( x - y )

= ( 5x - 4 ) ( x - y )

( x + y )³ + ( x - y )³

= 2x³ + 6xy²

= 2x ( x² + 3y² )

5 x^2 - 5xy + 4y - 4x 

= 5x ( x - y ) - 4 ( x - y ) 

= ( x - y ) ( 5x - 4 ) 

( x + y )^3 + ( x - y )^3 

= \(x^3+3x^2y+3xy^2+y^3+x^3-3x^2y+3xy^2-y^3\) 

= \(2x^3+6xy^2\) 

=\(2x\left(x^2+3y^2\right)\) 

a) \(5x\left(x-1\right)-3x\left(x-1\right)\)

\(=2x\left(x-1\right)\)

b) \(x\left(x+y\right)-5x-5y\)

\(=x\left(x+y\right)-5\left(x+y\right)\)

\(=\left(x-5\right)\left(x+y\right)\)

c) \(x\left(x-y\right)+y\left(y-x\right)\)

\(=\left(x-y\right)\left(x-y\right)\)

\(=\left(x-y\right)^2\)

d) \(x^2+xy+x=x\left(x+y+1\right)\)

a. 5x ( x - 1 ) - 3x ( x - 1 )

= ( 5x - 3x ) ( x - 1 )

b. x ( x + y ) - 5x - 5y

= x ( x + y ) - 5 ( x + y )

= ( x - 5 ) ( x + y )

c. x ( x - y ) + y ( y - x )

= x ( x - y ) - y ( x - y )

= ( x - y )²

 d. x² + xy + x 

= x ( x + y + 1 )

làm ơn giúp mk

- 6x² - 9xy + 15x

= - 3x ( 3y + 2x - 5 )

2x ( x - 3 ) + y ( x - 3 ) + ( 3 - x )

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

= ( 2x + y + 1 ) ( x - 3 )

:P

\(m^2+n^2=m+n+8\)

\(\Leftrightarrow4m^2-4m+1+4n^2-4n+1=34\)

\(\Leftrightarrow\left(2m-1\right)^2+\left(2n-1\right)^2=34\left(1\right)\)

Mà \(\left(2m-1\right)^2\ge0;\left(2n-1\right)^2\ge0;m,n\in N\)và \(5^2+3^2=3^2+5^2=34\)

Từ (1) suy ra 

\(\Leftrightarrow\hept{\begin{cases}2m-1=5\\2n-1=3\end{cases}}\Leftrightarrow\hept{\begin{cases}m=3\\n=2\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}2m-1=3\\2n-1=5\end{cases}}\Leftrightarrow\hept{\begin{cases}m=2\\n=3\end{cases}}\)

Vậy cặp số tự nhiên (m; n) thỏa mãn hệ thức \(m^2+n^2=m+n+8\)là \(\left\{\left(m=3;n=2\right);\left(m=2;n=3\right)\right\}\)

Ta có : \(m^2+n^2=m+n+8\)

\(\Leftrightarrow4m^2-4m+1+4n^2-4n+1=34\)

\(\Leftrightarrow\left(2m-1\right)^2+\left(2n-1\right)^2=34\left(1\right)\)

Mà \(\hept{\begin{cases}\left(2m-1\right)^2\ge0\\\left(2n-1\right)^2\ge0\end{cases}}\)và m , n thuộc N 

(1) \(\Rightarrow\left(2m-1\right)^2\le34\)

\(\Rightarrow2m-1\le5\Rightarrow2m\le6\Rightarrow m\le3\)

+) Khi m = 0 thì : \(m^2+n^2=m+n+8\) \(\Leftrightarrow n^2-n-8=0\)

\(\Delta=\left(-1\right)^2-4.\left(-8\right)=33\)\(\Rightarrow m\notin N\)

+) khi m= 1 thì : \(m^2+n^2=m+n+8\)\(\Leftrightarrow n^2-n-8=0\)

\(\Delta=\left(-1\right)^2-4.\left(-8\right)=33\)\(\Rightarrow m\notin N\)

+) Khi m =2 thì : \(m^2+n^2=m+n+8\)\(\Leftrightarrow n^2-n-6=0\)

\(\Delta=\left(-1\right)^2-4.\left(-6\right)=25>0\)

\(\Rightarrow\sqrt{\Delta}=\sqrt{25}=5\)  ; \(\hept{\begin{cases}n_1=\frac{1+5}{2}=3\left(TM\right)\\n_2=\frac{1-5}{2}=-2\left(L\right)\end{cases}}\)

+) Khi m = 3 thì : \(m^2+n^2=m+n+8\)\(\Leftrightarrow n^2-n-2=0\)

\(\Delta=\left(-1\right)^2-4.\left(-2\right)=9>0\)

\(\Rightarrow\sqrt{\Delta}=\sqrt{9}=3\); \(\hept{\begin{cases}n_3=\frac{1+3}{2}=2\left(TM\right)\\n_4=\frac{1-3}{2}=-1\left(L\right)\end{cases}}\)

vậy cặp snt ( m ; n ) thỏa mãn hệ thức \(m^2+n^2=m+n+8\)là \(\left(m;n\right)=\left(2;3\right)=\left(3;2\right)\)