B ir khoff-irregular boundary value problems for quadratic ordinary differential pencils of the second order have been considered. The spectral parameter may appear in a boundary condition, the equation contains an abstract linear operator while the boundary conditions contain internal points of an interval and a linear functional. Isomorphism and coerciveness with a defect are proved for such problems. Two-fold completeness of root functions of corresponding spectral problems is also established. As an application of the obtained results, an initial boundary value problem for second order parabolic equations is considered, and the well-posedness and completeness of the elementary solutions are proved.