Does the following constraint is convex or not? If it is not, how it can be implemented via the CVX?

\mathbf{x}^TA\mathbf{x} + C \leq(\mathbf{x}_1^T\mathbf{b})^2
where the vector \mathbf{x} = [\mathbf{x}_1^T, \mathbf{x}_2^T, ...,\mathbf{x}_N^T]^T \mathbf{x}_n is a vector and \mathbf{x}_n\geq 0 for n=1,2,...,N \mathbf{b}\geq 0 is a vector C\geq 0 is a constant

I will assume that x and x_1 are both optimization variables. Depending on the values of A and b, this is not necessarily convex. if you can consolidate the RHS into the LHS and end up with a convex quadratic on the LHS, then it will be convex.

If you do consolidate the RHS into what turns out to be a convex quadratic on the LHS, it would be convex quadratic <= -C
which would be infeasible unless C = 0.