We study how to compute global minimizers of the cubic-quartic regularization (CQR) problem. The CQR problem arises as a critical subproblem for getting efficient regularization methods for solving unconstrained nonlinear optimization. We propose a structured semidefinite programming (SDP) relaxation method for solving the CQR problem globally. For the CQR problem with n variables, the SDP relaxation has only three symmetric positive semidefinite matrix variables of sizes (n+1)-by-(n+1), 3-by-3 and 2-by-2 respectively. We show that our SDP relaxation is tight if and only if the sufficient global optimality condition holds for all nonzero global minimizers. In particular, if either the cubic regularization parameter is nonnegative, or the matrix that gives the quadratic term has a nonpositive eigenvalue, then the SDP relaxation is shown to be tight. Second, we show that all nonzero global minimizers have the same Euclidean norm for the tight case. Third, we give an algorithm to detect tightness and to obtain the set of all global minimizers. Numerical experiments demonstrate that our SDP relaxation method is both effective and computationally efficient. This gives a polynomial time algorithm for solving the CQR problem globally, under the sufficient global optimality condition.