a)\(\left(xy+1\right)^2-\left(x+y\right)^2\)
\(=\left[\left(xy+1\right)-\left(x+y\right)\right]\left(xy+1+x+y\right)\)
\(=\left(xy+1-x-y\right)\left(xy+1+x+y\right)\)
\(=\left(x-1\right)\left(x+1\right)\left(y-1\right)\left(y+1\right)\)
b)\(4a^2b^2-\left(a^2+b^2-c^2\right)^2\)
\(=\left(2ab\right)^2-\left(a^2+b^2-c^2\right)^2\)
\(=\left(2ab-a^2-b^2+c^2\right)\left(2ab+a^2+b^2-c^2\right)\)
\(=\left(a-b+c\right)\left(-a+b+c\right)\left(a+b-c\right)\left(a+b+c\right)\)
c)\(\left(a+b+c\right)^2+\left(a+b-c\right)^2\)
\(=a^2+b^2+c^2+2ab+2bc+2ca+a^2+b^2+c^2+2ab-2bc-2ac\)
\(=2a^2+2b^2+2c^2+4ab\)
\(=2\left(a^2+b^2+c^2+2ab\right)\)
d)\(x^3-7x-6\)
\(=x^3+3x^2+2x-3x^2-9x-6\)
\(=x\left(x^2+3x+2\right)-3\left(x^2+3x+2\right)\)
\(=\left(x^2+3x+2\right)\left(x-3\right)\)
\(=\left(x^2+x+2x+2\right)\left(x-3\right)\)
\(=\left[x\left(x+1\right)+2\left(x+1\right)\right]\left(x-3\right)\)
\(=\left(x+1\right)\left(x+2\right)\left(x-3\right)\)
