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Theory:

In the addition of algebraic expressions, while adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is the like term whose coefficient is the sum of the coefficients of these like terms.
Two ways to solve the addition of algebraic expressions:
• Horizontal Method.
• Column Method.
Horizontal Method: In this method, all expressions are written in a horizontal line, and then the terms are arranged to collect all the groups of like terms and then added.
Example:
Add $\phantom{\rule{0.147em}{0ex}}p\left(x\right)=\phantom{\rule{0.147em}{0ex}}2{x}^{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}6x\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5\phantom{\rule{0.147em}{0ex}}$ and $q\left(x\right)=\phantom{\rule{0.147em}{0ex}}3{x}^{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}2x\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}1$.

Add $\phantom{\rule{0.147em}{0ex}}p\left(x\right)=\phantom{\rule{0.147em}{0ex}}2{x}^{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}6x\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5\phantom{\rule{0.147em}{0ex}}$ and $q\left(x\right)=\phantom{\rule{0.147em}{0ex}}3{x}^{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}2x\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}1$.
$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}2{x}^{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}6x\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5\\ +3{x}^{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}2x\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}1\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\left(+\right)\left(-\right)\left(-\right)\\ \underset{¯}{\overline{5{x}^{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}4x\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}4}}\end{array}$