Answer
$$4x + 2y - z = 2$$
Work Step by Step
$$\eqalign{
& z = {x^2} + {y^2} + 3,{\text{ }}\left( {2,1,8} \right) \cr
& z - {x^2} - {y^2} - 3 = 0 \cr
& {\text{Considering }} \cr
& F\left( {x,y,z} \right) = z - {x^2} - {y^2} - 3 \cr
& {\text{Calculate the partial derivatives }} \cr
& {F_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {z - {x^2} - {y^2} - 3} \right] = - 2x \cr
& {F_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {z - {x^2} - {y^2} - 3} \right] = - 2y \cr
& {F_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {z - {x^2} - {y^2} - 3} \right] = 1 \cr
& {\text{At the point }}\left( {1,3,12} \right){\text{ the partial derivatives are}} \cr
& {F_x}\left( {2,1,8} \right) = - 2\left( 2 \right) = - 4 \cr
& {F_y}\left( {2,1,8} \right) = - 2\left( 1 \right) = - 2 \cr
& {F_z}\left( {2,1,8} \right) = 1 \cr
& {\text{An equation of the tangent plane at }}\left( {{x_0},{y_0},{z_0}} \right){\text{ is}} \cr
& {F_x}\left( {{x_0},{y_0},{z_0}} \right)\left( {x - {x_0}} \right) + {F_y}\left( {{x_0},{y_0},{z_0}} \right)\left( {y - {y_0}} \right) \cr
& + {F_z}\left( {{x_0},{y_0},{z_0}} \right)\left( {z - {z_0}} \right) = 0 \cr
& {\text{Substituting}} \cr
& {F_x}\left( {2,1,8} \right)\left( {x - 2} \right) + {F_y}\left( {2,1,8} \right)\left( {y - 1} \right) + {F_z}\left( {2,1,8} \right)\left( {z - 8} \right) = 0 \cr
& - 4\left( {x - 2} \right) - 2\left( {y - 1} \right) + \left( {z - 8} \right) = 0 \cr
& {\text{Simplifying}} \cr
& - 4x + 8 - 2y + 2 + z - 8 = 0 \cr
& - 4x - 2y + 2 + z = 0 \cr
& 4x + 2y - 2 - z = 0 \cr
& 4x + 2y - z = 2 \cr} $$