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1. \(\left(x+5\right)^3-x^3-125\)
\(=x^3+15x^2+75x+125-x^3-125\)
\(=15x^2+75x\)
2. \(x^3+6x^2+12x+8=0\)
\(\Leftrightarrow x^3+2x^2+4x^2+8x+4x+8=0\)
\(\Leftrightarrow x^2\left(x+2\right)+4x\left(x+2\right)+4\left(x+2\right)=0\)
\(\Leftrightarrow\left(x^2+4x+4\right)\left(x+2\right)=0\)
\(\Leftrightarrow\left(x+2\right)^2\left(x+2\right)=0\)
\(\Leftrightarrow\left(x+2\right)^3=0\)
\(\Leftrightarrow x+2=0\Leftrightarrow x=-2\)

\(\frac{x^3+125}{x^2-3x-40}=\frac{x^3+5^3}{\left(x^2+5x\right)-\left(8x+40\right)}=\frac{\left(x+5\right)\left(x^2-5x+25\right)}{x\left(x+5\right)-8\left(x+5\right)}\)
\(=\frac{\left(x+5\right)\left(x^2-5x+25\right)}{\left(x+5\right)\left(x-8\right)}=\frac{x^2-5x+25}{x-8}\)

\(27x^3-135x^2+225x-125\)
\(=\left(3x-5\right)^3\)
\(=\left(3.3-5\right)^3\)
\(=4^3=64\)

b: Ta có: \(\left(x+5\right)^3-x^3-125\)
\(=x^3+15x^2+75x+125-x^3-125\)
\(=15x^2+75x\)

\(\left(x+5\right)^3-x^3-125\)
\(=x^3+15x^2+15x+125-x^3-125\)
\(=15x^2+15x=15x.\left(x+1\right)\)
Chúc bạn học tốt!!!
\(\left(x+5\right)^3-x^3-125=\left(x+5-x\right)\left[\left(x+5\right)^2+x\left(x+5\right)+x^2\right]\)
\(=5\left(3x^2+15x+25\right)\)

Lời giải :
1. \(\left(\frac{1}{2}a+b\right)^3+\left(\frac{1}{2}a-b\right)^3\)
\(=\frac{a^3}{8}+\frac{3a^2b}{4}+\frac{3ab^2}{2}+b^3+\frac{a^3}{8}-\frac{3a^2b}{4}+\frac{3ab^2}{2}-b^3\)
\(=\frac{a^3}{4}+3ab^2\)
Lời giải :
2. \(x^3-3x^2+3x-1=0\)
\(\Leftrightarrow\left(x-1\right)^3=0\)
\(\Leftrightarrow x-1=0\)
\(\Leftrightarrow x=1\)
Vậy...
\(\left(x+5\right)^3-x^3-125\)
\(=x^3+5x^2+25x+125-x^3-125\)
\(=5x^2+25x\)