Have you considered checking out the snippet below?
Feel free to customize the rest of the styling on your own.
.container {
padding-top: 50px;
}
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.5/css/bootstrap.min.css" />
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
<script src="https://code.jquery.com/jquery-1.11.3.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.5/js/bootstrap.min.js"></script>
<div class="container">
<div class="row">
<div class="col-xs-12">
<div class="search-box">
<form class="form-inline">
<div class="form-group">
<div class="input-group">
<input type="text" class="form-control" aria-label="..." placeholder="Search ...">
<div class="input-group-btn">
<button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">All categories <span class="caret"></span>
</button>
<ul class="dropdown-menu dropdown-menu-right">
<li><a href="#">All categories</a>
</li>
<li><a href="#">Another action</a>
</li>
<li><a href="#">Something else here</a>
</li>
<li role="separator" class="divider"></li>
<li><a href="#">Separated link</a>
</li>
</ul>
<button type="submit" class="btn btn-default">
<i class="fa fa-search"></i>
</button>
</div>
<!-- /btn-group -->
</div>
<!-- /input-group -->
</div>
</form>
</div>
</div>
</div>
</div>