a)\(x^2+3x+7\)

\(=x^2+3x+\dfrac{9}{4}+\dfrac{19}{4}\)

\(=\left(x+\dfrac{3}{2}\right)^2+\dfrac{19}{4}\ge\dfrac{19}{4}\)

When \(x=-\dfrac{3}{2}\)

b)\(2x^2+9y^2-6xy-6x-12y+2004\)

\(=\left(x^2-6xy+4x+9y^2-12y+4\right)+\left(x^2-10x+25\right)+1975\)

\(=\left(x-3y+2\right)^2+\left(x-5\right)^2+1975\ge1975\)

When \(x=5;y=\dfrac{7}{3}\)