Ta có:
\(VT=\left(a-1\right)\left(a-2\right)\left(a+3\right)-\left(a+1\right)\left(a+2\right)\left(a-3\right)\)
\(=\left[a\left(a-2\right)-\left(a-2\right)\right]\left(a+3\right)-\left[a\left(a+2\right)+\left(a+2\right)\right]\left(a-3\right)\)
\(=\left[\left(a^2-2a\right)-\left(a-2\right)\right]\left(a+3\right)-\left[\left(a^2+2a\right)+\left(a+2\right)\right]\left(a-3\right)\)
\(=\left[a^2-2a-a+2\right]\left(a+3\right)-\left[a^2+2a+a+2\right]\left(a-3\right)\)
\(=\left[a^2-3a+2\right]\left(a+3\right)-\left[a^2+3a+2\right]\left(a-3\right)\)
\(=\left[a^2\left(a+3\right)-3a\left(a+3\right)+2\left(a+3\right)\right]-\left[a^2\left(a-3\right)+3a\left(a-3\right)+2\left(a-3\right)\right]\)
\(=\left[\left(a^3+3a^2\right)-\left(3a^2+9a\right)+\left(2a+6\right)\right]-\left[\left(a^3-3a^2\right)+\left(3a^2-9a\right)+\left(2a-6\right)\right]\)
\(=\left[a^3+3a^2-3a^2-9a+2a+6\right]-\left[a^3-3a^2+3a^2-9a+2a-6\right]\)
\(=\left[a^3-7a+6\right]-\left[a^3-7a-6\right]\)
\(=a^3-7a+6-a^3+7a+6\)
\(=12\) (đpcm)
Áp dụng kết quả trên, ta có:
\(149.148.153-151.152.147\)
\(=\left(150-1\right)\left(150-2\right)\left(150+3\right)-\left(150+1\right)\left(150+2\right)\left(150-3\right)\)
\(=12\)
\(99.98.103-101.102.97\)
\(=\left(100-1\right)\left(100-2\right)\left(100+3\right)-\left(100+1\right)\left(100+2\right)\left(100-3\right)\)
\(=12\)
Do đó: \(149.148.153-151.152.147=99.98.103-101.102.97\) (đpcm)
