This is a simple splash screen created without using any images and using Linear gradients for background and Text widget for logo to demonstrate basic of flutter.

main.dart

import 'package:flutter/material.dart';
import 'package:flutter_quiz/simple_splash.dart';

void main() => runApp(new MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new MaterialApp(
      title: 'Flutter Demo',
      theme: new ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: new SplashScreen(),
    );
  }
}

simple_splash.dart

import 'package:flutter/material.dart';

class SplashScreen extends StatelessWidget {
  Widget _getLogo() {
    return Padding(
      padding: const EdgeInsets.symmetric(vertical: 30.0),
      child: Column(
        children: <Widget>[
          Row(
            mainAxisAlignment: MainAxisAlignment.center,
            children: <Widget>[
              Text(
                'Flutter',
                style: TextStyle(
                  color: Colors.white,
                  fontSize: 50.0,
                  fontFamily: 'Kaushan',
                ),
              ),
              Text(
                ' Quiz',
                style: TextStyle(
                  color: Colors.white,
                  fontSize: 30.0,
                  fontFamily: 'Monda',
                ),
              ),
            ],
          ),
          Text(
            'Learn from scoring',
            textScaleFactor: 0.8,
            style: TextStyle(
                color: Colors.white,
                fontSize: 20.0,
                fontFamily: 'MavenPro',
                letterSpacing: 2.0,
                wordSpacing: 3.0),
          ),
        ],
      ),
    );
  }

  Widget _getDescription() {
    return Padding(
      padding: const EdgeInsets.all(30.0),
      child: Text(
        'The fun app to learn Flutter from answering to questions. The Flutter framework for to build cross platform high quality native apps using single code base.',
        textAlign: TextAlign.justify,
        style: TextStyle(
          color: Colors.white,
          fontSize: 17.0,
          fontFamily: 'MovenPro',
        ),
      ),
    );
  }

  Material _getStartedButton() {
    return Material(
      color: Colors.white,
      borderRadius: BorderRadius.circular(30.0),
      child: InkWell(
        splashColor: Colors.black12,
        highlightColor: Colors.lightBlueAccent,
        onTap: () {
          print('go to start');
        },
        child: Container(
          padding: const EdgeInsets.symmetric(vertical: 12.0, horizontal: 30.0),
          child: Text(
            'Get Started',
            style: TextStyle(color: Colors.black, fontSize: 22.0, fontFamily: 'MovenPro'),
          ),
        ),
      ),
    );
  }

  Widget _getHelp() {
    return Padding(
      padding: const EdgeInsets.all(10.0),
      child: GestureDetector(
        onTap: () {
          print('help me');
        },
        child: Text(
          'Help',
          style: TextStyle(color: Colors.white, fontFamily: 'MovenPro', fontSize: 18.0),
        ),
      ),
    );
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Stack(
        children: <Widget>[
          SplashBackground(),
          SafeArea(
            child: Column(
              mainAxisAlignment: MainAxisAlignment.spaceBetween,
              crossAxisAlignment: CrossAxisAlignment.center,
              children: <Widget>[
                _getLogo(),
                _getDescription(),
                _getStartedButton(),
                _getHelp(),
              ],
            ),
          ),
        ],
      ),
    );
  }
}

class SplashBackground extends StatelessWidget {
  Container _getFirstGradient() {
    return new Container(
      // margin: new EdgeInsets.only(top: 190.0),
      // height: 110.0,
      decoration: new BoxDecoration(
        gradient: new LinearGradient(
          colors: <Color>[new Color(0x001E88E5), new Color(0xFF1E88E5)],
          stops: [0.0, 0.5],
          begin: const FractionalOffset(0.0, 0.0),
          end: const FractionalOffset(0.0, 1.0),
        ),
      ),
    );
  }

  Container _getSecondGradient() {
    return new Container(
      // margin: new EdgeInsets.only(top: 190.0),
      // height: 110.0,
      decoration: new BoxDecoration(
        gradient: new LinearGradient(
          colors: <Color>[new Color(0x00039BE5), new Color(0xFF039BE5)],
          stops: [0.0, 0.5],
          begin: const FractionalOffset(0.0, 1.0),
          end: const FractionalOffset(0.0, 0.0),
        ),
      ),
    );
  }

  @override
  Widget build(BuildContext context) {
    return Stack(
      children: <Widget>[
        _getFirstGradient(),
        _getSecondGradient(),
      ],
    );
  }
}