a.

\(x^2\left(x+5\right)-9x=45\\ \Leftrightarrow x^2\left(x+5\right)-9\left(x+5\right)=0\\ \Leftrightarrow\left(x+5\right)\left(x^2-9\right)=0\\ \Leftrightarrow\left(x+5\right)\left(x-3\right)\left(x+3\right)=0\\ \)

\(\Rightarrow\left[{}\begin{matrix}x+5=0\\x+3=0\\x-3=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=-5\\x=-3\\x=3\end{matrix}\right.\)

b.

\(9\left(5-x\right)+x^2-10x=-25\\ \Leftrightarrow9\left(5-x\right)+x^2-10x+25=0\\ \Leftrightarrow9\left(5-x\right)+\left(x-5\right)^2=0\\ \Leftrightarrow9\left(5-x\right)+\left(5-x\right)^2=0\\ \Leftrightarrow\left(5-x\right)\left(9+5-x\right)=0\\ \Leftrightarrow\left(5-x\right)\left(14-x\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}5-x=0\\14-x=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=5\\x=14\end{matrix}\right.\)

Các bài còn lại đều đặt thừa số chung tương tự như 2 câu trên.

   x⁴ - x³ - x + 1

= x³.(x-1) - ( x -1)

= (x-1).(x³ -1)

=(x-1)(x-1)(x²+x+1)

9.(x-5)²  -( x-7)²

={ 3.(x-5)}² - (x-7)²

= [ 3.(x-5) - x + 7][3.(x-5) + x-7]

= (3x -15 - x + 7)( 3x - 15 + x -7)

= (2x -8)(4x - 22)

    (x² + 2x - y² + 1) : (x + y + 1)

= { (x² + 2x + 1) -y²) : (x +y +1)

= { (x +1)² -y² } : (x + y+1)

= ( x + 1 - y)( x +1+y) : (x+ 1+y)

= x + 1 - y

Ta có:

\(\left(a-b\right)^2\ge0\\ \Leftrightarrow a^2-2ab+b^2\ge0\\ \Rightarrow a^2+b^2\ge2ab\)

Dấu "=" xảy ra khi \(a=b\)

Tương tự ta cũng có:

\(b^2+c^2\ge2bc\) . Dấu "=" xảy ra khi \(b=c\)

\(c^2+a^2\ge2ca\) . Dấu "=" xảy ra khi \(c=a\)

Suy ra: 

\(2\left(a^2+b^2+c^2\right)\ge2\left(ab+bc+ca\right)\\ \Leftrightarrow a^2+b^2+c^2\ge ab+bc+ca\)

Dấu "=" xảy ra khi \(a=b=c\) . đpcm