Minimum Value

We expand the given expression to obtain

((x^2 + 6x + 9) + 2(y^2 - 4y + 4) + 4(x^2 - 14x + 49) + (y^2 + 8y + 16))

We simplify to obtain

$5x^2 - 50x + 3y^2 + 229$

We remove a common factor of 5 from the first two terms

$5(x^2 - 10x) + 3y^2 + 229$

and then complete the square to obtain

$5(x^2 - 10x + 125) - 125 +3y^2 + 229$

This gives

$5(x - 5)^2 + 3y^2 + 104$

Since $(x - 5)^2 ≥ 0$ for all real numbers $x$ and $3y^2 ≥ 0$ for all real numbers $y$ , then the minimum value of $5(x - 5)^2 + 3y^2 + 104$
is 5(0) + 3(0) + 104 = 104