<p><span style="color:#27ae60;"><span style="font-size:20px;">Solving the Problem</span></span></p>
<p>This question is best solved by <strong><span style="color:#8e44ad;">Having a System</span></strong>. We have to figure out which equation will give an even result for both even and odd values of $x$. We can just choose two simple numbers for this:</p>
<table border="1" cellpadding="1" cellspacing="1">
<thead>
<tr>
<th scope="row">Equation</th>
<th scope="col">Result when x is 1 (Odd)</th>
<th scope="col">Result when x is 0 (Even)</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">$x^2-x-1$</th>
<td>$1^2-1-1=-1$</td>
<td>No need to calculate</td>
</tr>
<tr>
<th scope="row">$x^2-4x+6$</th>
<td>$1^2-4(1)+6=3$</td>
<td>No need to calculate</td>
</tr>
<tr>
<th scope="row">$x^2-5x+5$</th>
<td>$1^2-5(1)+5=1$</td>
<td>No need to calculate</td>
</tr>
<tr>
<th scope="row">$x^2+3x+8$</th>
<td>$1^2+3(1)+8=12$</td>
<td>$0^2+3(0)+8=8$</td>
</tr>
<tr>
<th scope="row">$x^2+2x+10$</th>
<td>$1^2+2(1)+10=13$</td>
<td>No need to calculate</td>
</tr>
</tbody>
</table>
<p>We can see that if $x=1$, only $x^2+3x+8$ gives us an even value, so we can eliminate everything else.</p>
<p>So, the correct answer would be <span style="color:#27ae60;">D</span>.</p>