Simplifying boolean equations with K-map
Here's the initial k-map:
$$
\begin{array}{rl}
\begin{smallmatrix}\begin{array}{r|cccc}
&amp;\overline{A}\:\overline{B}&amp;\overline{A}\: B&amp;A\: B&amp;A\: \overline{B}\\
\hline
\overline{C}\:\overline{D}&amp;1&amp;0&amp;1&amp;1\\
\overline{C}\:D&amp;0&amp;0&amp;1&amp;1\\
C\: D&amp;0&amp;0&amp;0&amp;1\\
C\:\overline{D}&amp;0&amp;0&amp;1&amp;1
\end{array}\end{smallmatrix}
\end{array}
$$
And here's how I circled things (using Paint):

A brute force algebra method might go like this:
$$\begin{align*}
&amp; A\,B\,C\,\overline{D} + A\left(\overline{B} + \overline{C} + \overline{D}\right) + \overline{A}\,\overline{B}\,\overline{C}\,\overline{D}\\\\
&amp; A\,B\,C\,\overline{D} + \overline{A}\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B} + A\,\overline{C} + A\,\overline{D}\\\\
&amp; A\,B\,C\,\overline{D} + \overline{A}\,\overline{B}\,\overline{C}\,\overline{D}\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,\overline{C}\,D + A\,\overline{B}\,C\,\overline{D} + A\,\overline{B}\,C\,D\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,\overline{C}\,D + A\,B\,\overline{C}\,\overline{D} + A\,B\,\overline{C}\,D\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,C\,\overline{D} + A\,B\,\overline{C}\,\overline{D} + A\,B\,C\,\overline{D}\\\\
&amp; A\,B\,C\,\overline{D} + \left(\overline{A}\,\overline{B}\,\overline{C}\,\overline{D}\right.\\&amp;\quad\quad\quad\quad\quad\quad + \left.A\,\overline{B}\,\overline{C}\,\overline{D}\right) + A\,\overline{B}\,\overline{C}\,D + A\,\overline{B}\,C\,\overline{D} + A\,\overline{B}\,C\,D\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,\overline{C}\,D + A\,B\,\overline{C}\,\overline{D} + A\,B\,\overline{C}\,D\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,C\,\overline{D} + A\,B\,\overline{C}\,\overline{D} + A\,B\,C\,\overline{D}\\\\
&amp; \overline{B}\,\overline{C}\,\overline{D}\\&amp;\quad\quad\quad\quad\quad\quad + A\,B\,C\,\overline{D} + A\,\overline{B}\,\overline{C}\,D + A\,\overline{B}\,C\,\overline{D} + A\,\overline{B}\,C\,D\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,\overline{C}\,D + A\,B\,\overline{C}\,\overline{D} + A\,B\,\overline{C}\,D\\&amp;\quad\quad\quad\quad\quad\quad + A\,\overline{B}\,\overline{C}\,\overline{D} + A\,\overline{B}\,C\,\overline{D} + A\,B\,\overline{C}\,\overline{D} + A\,B\,C\,\overline{D}\\\\
&amp; \overline{B}\,\overline{C}\,\overline{D} + A\left(\overline{B}\left[\overline{C}\,\overline{D} + \overline{C}\,D + C\,\overline{D} + C\,D\right] + B\left[\overline{C}\,\overline{D} + \overline{C}\,D + C\,\overline{D}\right]\right)\\\\
&amp; \overline{B}\,\overline{C}\,\overline{D} + A\left(\overline{B} + B\left[\overline{C}\,\overline{D} + \overline{C}\,D + C\,\overline{D}\right]\right)\\\\
&amp; \overline{B}\,\overline{C}\,\overline{D} + A\left(\overline{B} + \overline{C}\,\overline{D} + \overline{C}\,D + C\,\overline{D}\right)\\\\
&amp; \overline{B}\,\overline{C}\,\overline{D} + A\left(\overline{B} + \overline{C} + C\,\overline{D}\right)\\\\
&amp; \overline{B}\,\overline{C}\,\overline{D} + A\left(\overline{B} + \overline{C} + \overline{D}\right)
\end{align*}$$
I hope you follow those steps okay.