title: How is f(x) = 4x^2 - 5x + 3 is O(x^2) derived
body: <p>Here are the steps that are used to prove the above</p>
  
  <pre><code>|f(x)| = |4x^2 – 5x + 3|
  &lt;= |4x^2|+ |- 5x| + |3|
  &lt;= 4x^2 + 5x + 3, for all x &gt; 0
  &lt;= 4x^2 + 5x^2 + 3x^2, for all x &gt; 1
  &lt;= 12x^2, for all x &gt; 1 
  </code></pre>
  
  <p>Hence we conclude that f(x) is O(x^2)</p>
  
  <p>I <a href="https://stackoverflow.com/questions/14249151/big-o-notation-finding-c-and-n0">referred this</a> But it does not help </p>
  
  <p>Can someone explain the above proof step by step?</p>
  
  <ol>
  <li>Why the absolute value of f(x) is taken ?</li>
  <li>Why and how were all the term replaced by x^2 term?</li>
  </ol>