a/ \(\left(x+2\right)\left(x+5\right)\left(x+3\right)\left(x+4\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=\left(x^2+7x+10\right)^2+2\left(x^2+7x+10\right)-24\)
\(=\left(x^2+7x+10-4\right)\left(x^2+7x+10+6\right)\)
\(=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
b/ \(\left(x^2-2x+4\right)\left(x^2+3x+4\right)-14x^2\)
\(=x^2\left[\left(x-2+\frac{4}{x}\right)\left(x+3+\frac{4}{x}\right)-14\right]\)
\(=x^2\left[\left(x-2+\frac{4}{x}\right)^2+5\left(x-2+\frac{4}{x}\right)-14\right]\)
\(=x^2\left(x-2+\frac{4}{x}-2\right)\left(x-2+\frac{4}{x}+7\right)\)
\(=x^2\left(x-4+\frac{4}{x}\right)\left(x+5+\frac{4}{x}\right)\)
\(=\left(x^2-4x+4\right)\left(x^2+5x+4\right)\)
\(=\left(x-2\right)^2\left(x+1\right)\left(x+4\right)\)
c/ \(a^2b-a^2c+b^2\left(a-c\right)+c^2a-c^2b\)
\(=b\left(a^2-c^2\right)-ac\left(a-c\right)+b^2\left(a-c\right)\)
\(=\left(a-c\right)\left(ab+bc-ac+b^2\right)\)
d/ \(a^3+b^3+3ab\left(a+b\right)+c^3-3ab\left(a+b\right)-3abc\)
\(=\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(\left(a+b\right)^2-\left(a+b\right)c+c^2\right)-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-ac-bc\right)\)
