Extend or set up the ladder so you can comfortably reach the downspout.
Pull the downspout away from the gutter if it is a slip connection. If it is glued, you will need to use the hacksaw or reciprocating saw.
Install the new PVC coupler to connect the gutter to the new downspout. The new connection should be swabbed with PVC cleaner and adhered with the PVC glue. The coupler should be straight if going straight down with the new pipe, or can be angled. A 45-degree angle should be used instead of a 90-degree angle to reduce the chance of blockage.
Install the new pipe to the coupler with the PVC cleaner and glue. Fasten each length of pipe to the side of the house with the gutter bracket and either the nails or screws depending on the house's siding material.
Install a 45-degree coupler toward the base of the downspout pipe. Use the same cleaner and glue as before. Attach the desired length of pipe to extend the drainage from the roof. Use angled couplers as needed to direct the flow. Attach all fittings with cleaner and glue.
Locate the existing end of the downspout. It might be necessary to dig around it. Determine whether a new connection can be made with the existing pipework, or use the hacksaw or reciprocating saw to cut it to receive new components.
Attach, with PVC cleaner and glue, a new coupler to connect the existing downspout to the new pipe extensions.
Lay the pipe sections to the desired length, attaching each with the cleaner and glue. Use the angled fittings to divert the flow away from the house.