\(B^2=x^2\left(y^2+1\right)+y^2\left(x^2+1\right)+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}\)
\(=2x^2y^2+x^2+y^2+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}\)
\(=x^2y^2+\left(x^2y^2+x^2+y^2+1\right)+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}-1\)
\(=x^2y^2+\left(x^2+1\right)\left(y^2+1\right)+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}-1\)
\(=\left(xy+\sqrt{\left(x^2+1\right)\left(y^2+1\right)}\right)^2-1\)
\(=\left(\sqrt{2008}\right)^2-1=2007\)
\(\Rightarrow B=\sqrt{2007}\)
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