Determine if the Set is or not a vector subspace

not really solid in determining vector subspace and now i got handed this crazy problem. please help i appreciate any effort.

W={[x1,x2,x3,x4] : x1^2+4x2^2+x3^2-4x1x2+2x1x3-4x2x3=0 and 2x1+3x2+4x3=0}

We need to write x1^2 + 4x2^2 + x3^2 - 4x1x2 + 2x1x3 - 4x2x3 as a perfect square first, to have any real hope of solving this problem.

Observe that

x1^2 + 4x2^2 + x3^2 - 4x1x2 + 2x1x3 - 4x2x3
= x1^2 + (-2x2)^2 + x3^2 + 2(x1)(-2x2) + 2x1x3 + 2(-2x2)(x3)
= (x1 - 2x2 + x3)^2.

Therefore, x1^2 + 4x2^2 + x3^2 - 4x1x2 + 2x1x3 - 4x2x3 = 0 is equivalent to
x1 - 2x2 + x3 = 0.

So we can rewrite W as {[x1,x2,x3,x4] : x1-2x2+x3=0 and 2x1+3x2+4x3=0}.

The solution set of a system of linear equations, with zeros for all the constant terms, is always a subspace. Since W is the solution set of a system of linear equations, with zeros for all the constant terms, W is a subspace.

Lord bless you today!