a) \(\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)+1\)
\(=\left(x^2+9x+18\right)\left(x^2+9x+20\right)+1\)
\(=\left(x^2+9x+19\right)^2-1+1\)
\(=\left(x^2+9x+19\right)^2\)
b) \(x^2+y^2+2x+2y+2\left(x+1\right)\left(y+1\right)+2\)
\(=\left(x^2+2x+1\right)+\left(y^2+2y+1\right)+2\left(x+1\right)\left(y+1\right)\)
\(=\left(x+1\right)^2+\left(y+1\right)^2+2\left(x+1\right)\left(y+1\right)\)
\(=\left(x+1+y+1\right)^2=\left(x+y+2\right)^2\)
c) \(x^2-2x\left(y+2\right)+y^2+4y+4\)
\(=x^2-2x\left(y+2\right)+\left(y^2+4y+4\right)\)
\(=x^2-2x\left(y+2\right)+\left(y+2\right)^2\)
\(\left(x-y-2\right)^2\)
d) \(x^2+2x\left(y+1\right)+y^2+2y+1\)
\(=x^2+2x\left(y+1\right)+\left(y^2+2y+1\right)\)
\(=x^2+2x\left(y+1\right)+\left(y+1\right)^2\)
\(=\left(x+y+1\right)^2\)
