Place the 52-inch plywood sides on a worktable. Measure from one end (the top) and make marks at 36 and 36 3/4 inches. Place the framing square on the marks and draw lines across the material.
Position the sides so the long edges are touching and the ends are even. Mark the adjacent edges as the back. Put the 15 1/4-inch pieces of lumber (cleats) on the bottom edge of the sides and the back edges. Secure the cleats to the sides with the 3-penny box nails.
Put the 36-inch plywood (shelves) on the worktable. Measure from one 16-inch side and make marks at 17 5/8 and 18 3/8 on two of the shelves. Glue the 15 1/4-inch edges of the 11-by-15 1/4-inch plywood (divider) and secure it between the lines on the two shelves with the finish nails.
Apply glue to one long edge of three of the 36-inch lumber (stabilizers). Secure them to the back edge of the shelves with the finish nails. The stabilizers on the bottom shelves face up while the one on the top shelf faces down. Apply glue to the ends of the shelves and secure them between the sides with the finish nails.
Lay the cabinet on its front. Apply glue to the exposed edges and secure the 37 1/2-by-48 1/2-by-1/4-inch plywood (back) to the cabinet with the 2-penny box nails. Turn the cabinet over and clean up the glue.
Install the remaining 36-inch piece of lumber on the front of the cleats with the finish nails. Drive one or two nails though the bottom shelf into the piece.
Cover the plywood edges of the cabinet and the 17 13/16-by-35 3/4-inch plywood doors with the veneer edge band. Cut strips of the edge band 2 inches longer than the edge they will cover with a utility knife. Hold the strip against the edge while heating it with a hot iron until the piece sticks to the edge.
Rub the edge band with a small block of wood to force the glue into the pores of the plywood. Fold the ends of the strip over the corner of the plywood to break it off. Sand the long edges with a sanding block and 100-grit sandpaper. Install the pocket door slides and the TV pull-out slide according to the manufacture's instructions.