php mail() cannot auto-send email to @hotmail.com, @outlook.com

Question

I use a very basic and simple mail() php function to automatically send email to the new users so that they can activate their account, which looks like this:

    $body = "Thank you for your registering at ". BASE_URL. ". To activate your account, please click on this link:\n\n";
    $body .= BASE_URL . 'activate.php?e=' . urlencode($e) . "&a=$a";
    mail($trimmed['email'], 'Registration Confirmation', $body, 'From: admin@hiteachers.com');  

In which the $trimmed['email'] is the email the new user uses to enter the website registration form after validation, and the admin@hiteachers.com is my REAL email address, which I use with Google apps (I followed the link https://support.google.com/a/answer/33353?hl=en (#2,3 and 4) to edit the MX records in my domain manager).

Note : - I bought the domain and a shared linux server at Godaddy.

• I also consulted PHP mail() function cannot send to hotmail?, and some other related links, but it didn't help.

• I also followed the link https://mail.live.com/mail/troubleshooting.aspx to troubleshoot the SMTP Non-Delivery Report by adding hiteachers.com IN TXT "v=spf1 -all" into the TXT records and waited for 48 hours, but no luck!

• I also added the TXT value v=spf1 a mx -all into my domain TXT records as per other stackoverflow users' suggestions, and waited for 48 hours, but no luck!

• I also added the from email address to the safe contact list of the recipient addresses at @hotmail.com, and @outlook.com, which i used for testing, but no luck!

• No email has been sent to @hotmail or @outlook inbox, even in the junk folder, and no bouce-back email either (because the recipient accounts are real).

• It works well with real gmail and yahoo email recipient addresses.

• I tried sending email directly from admin@hiteachers.com to the real recipient email address at outlook.com, it stayed there in the inbox (means that it accepts my admin@hiteachers.com address). However, when I viewed the message source, it says This sender failed our fraud detection checks and may not be who they appear to be.

Do you know what was wrong with it, or have you ever been in the same situation as mine? And the important part of my question is what I should do to eliminate the whole issue, please?

UPDATED

After adding the SPF and other required records, which you can take a look at here http://mxtoolbox.com/ for the domain hiteachers.com, I received the report from <email>noreply-dmarc-support@google.com</email> like this, and all my testing auto-email to @yahoo.com and @gmail.com are going to spam folders, but not the inbox before:

<?xml version="1.0" encoding="UTF-8" ?>
<feedback>
  <report_metadata>
    <org_name>google.com</org_name>
    <email>noreply-dmarc-support@google.com</email>
    <extra_contact_info>https://support.google.com/a/answer/2466580</extra_contact_info>
    <report_id>10844434555482221094</report_id>
    <date_range>
      <begin>1458172800</begin>
      <end>1458259199</end>
</date_range>
  </report_metadata>
  <policy_published>
    <domain>hiteachers.com</domain>
    <adkim>r</adkim>
    <aspf>r</aspf>
    <p>quarantine</p>
    <sp>quarantine</sp>
    <pct>100</pct>
  </policy_published>
  <record>
    <row>
      <count>1</count>
      <policy_evaluated>
        <disposition>quarantine</disposition>
        <dkim>fail</dkim>
        <spf>fail</spf>
      </policy_evaluated>
    </row>
    <identifiers>
      <header_from>hiteachers.com</header_from>
    </identifiers>
    <auth_results>
      <spf>
        <domain>xxxxxxxxx.xxx.xxx.xxxx.secureserver.net</domain>
        <result>none</result>
      </spf>
    </auth_results>
  </record>
</feedback>