5 x^2+x+3y^2 +2y +z^2+2z-8xy-8yz=3

5(x^2+x/5) +3(y^2 +2/3y)+(z^2 +2z)-8x(y+z)-3=0

We will convert the first three brackets perfect square by adding and subtracting 1/100, 1/9 and 1 respectively inside the brackets

5(x^2+x/5 +1/100 -1/100)+3(y^2+ 2/3y + 1/9 - 1/9)+(z^2 +2z +1 -1) -8x(y+z) -3=0

5(x+1/10)^2 - 5/100 +3(y+ 1/3 )^2 -3/9 +(z+1)^2 - 1 -8x(y+z) -3 =0

5(x+1/10)^2 +3(y+1/3)^2 +(z+1)^2 -8x(y+z) - 263/60 =0