Роуты:

Route::get('/', 'MainController@index');
Route::get('/registration', 'MainController@registration')->name('registration');

Route::get('home', 'TasksController@home');
Route::get('home/create', 'TasksController@create')->name('create');

Route::post('home/store', 'TasksController@store')->name('tasks.store');

Контроллер:

namespace App\Http\Controllers;

use Illuminate\Http\Request;
use App\Task;

class TasksController extends Controller
{
	public function index() {
		return view('index');
	}
	public function home() {
		return view('tasks.home');
	}
	public function create() {
		return view('tasks.create');
	}
	public function store(Request $request) {
		$task = new Task;
		$task->title = $request->get('title');
		$task->description = $request->get('description');

		$task->save();

		return redirect()->route('tasks.home');
	}
}

Форма:

@extends('layout')

@section('content')

<div class="container">
	<h3>Добавить запись</h3>
	<div class="row">
		<div class="col-md-12">
			{!! Form::open(['route' => ['tasks.store']]) !!}
			<div class="form-group">
				<input type="text" class="form-control" name="title">
				<br>
				<textarea name="description" id="" cols="30" rows="10" class="form-control"></textarea>
				<br>
				<button class="btn btn-success">Добавить</button>
			</div>
			{!! Form::close() !!}
		</div>
	</div>
</div>

@endsection