The following output looks so dense. Could you make it more loose in both vertical and horizontal?

MWE

documentclass[preview,border=12pt,12pt]{standalone}

usepackage{amsmath,xcolor}

lettempunderbrace

defunderbrace#1_#2{{color{red}temp{{color{black}#1}}_{#2}}}



begin{document}

$displaystyle

begin{aligned}

x^2+6x+10 

&= underbrace{x^2}_{a^2} underbrace{+2x (3)}_{+2ab} + 10\

&= underbrace{x^2}_{a^2} underbrace{+2x (3)}_{+2ab} underbrace{+(3)^2 -(3)^2}_{+b^2-b^2} +10\

&= underbrace{x^2+2x(3)+(3)^2}_{a^2+2ab+b^2}  -(3)^2 +10\

&= underbrace{(x+3)^2}_{(a+b)^2} -9 +10\

&= (x+3)^2+1

end{aligned}

$

end{document}

Any constructive suggestions, advice, ideas, etc are always welcome!

I'd use mathstrut to cover the most common cases avoiding phantoms; also textcolor is better than color in math mode.

You need to add {} in front of + to get decent spacing. I also added a couple of thin spaces for separating the braces.

documentclass[border=12pt,12pt]{standalone}

usepackage{amsmath,xcolor}



letlatexunderbraceunderbrace

defunderbrace#1_#2{%

  {,textcolor{red}{latexunderbrace{textcolor{black}{mathstrut#1}}_{#2}},}%

}



begin{document}

$displaystyle

begin{aligned}

x^2+6x+10 

&= underbrace{x^2}_{a^2} underbrace{{}+2x (3)}_{+2ab} + 10\

&= underbrace{x^2}_{a^2} underbrace{{}+2x (3)}_{+2ab} underbrace{+(3)^2 -(3)^2}_{+b^2-b^2} +10\

&= underbrace{x^2+2x(3)+(3)^2}_{a^2+2ab+b^2}  -(3)^2 +10\

&= underbrace{(x+3)^2}_{(a+b)^2} -9 +10\

&= (x+3)^2+1

end{aligned}

$

end{document}

A slightly better version where the space in front of the + is removed

documentclass[border=12pt,12pt]{standalone}

usepackage{amsmath,xcolor}



letlatexunderbraceunderbrace

defunderbrace#1_#2{%

  {,textcolor{red}{latexunderbrace{textcolor{black}{mathstrut#1}}_{#2}},}%

}

newcommand{hplus}{{mspace{-medmuskip}}+}



begin{document}

$displaystyle

begin{aligned}

x^2+6x+10 

&= underbrace{x^2}_{a^2} underbrace{hplus 2x (3)}_{+2ab} + 10\

&= underbrace{x^2}_{a^2} underbrace{hplus2x (3)}_{+2ab} underbrace{+(3)^2 -(3)^2}_{+b^2-b^2} +10\

&= underbrace{x^2+2x(3)+(3)^2}_{a^2+2ab+b^2}  -(3)^2 +10\

&= underbrace{(x+3)^2}_{(a+b)^2} -9 +10\

&= (x+3)^2+1

end{aligned}

$

end{document}

documentclass[preview,border=12pt,12pt]{standalone}

usepackage{amsmath,xcolor}

lettempunderbrace

defunderbrace#1_#2{{color{red}temp{{color{black}{strut}#1{}}}_{#2}}}



begin{document}

$displaystyle

openup1jot

begin{aligned}

x^2+6x+10 

&= underbrace{x^2}_{a^2} underbrace{+2x (3)}_{+2ab} + 10\

&= underbrace{x^2}_{a^2} underbrace{+2x (3)}_{+2ab} underbrace{+(3)^2 -(3)^2}_{+b^2-b^2} +10\

&= underbrace{x^2+2x(3)+(3)^2}_{a^2+2ab+b^2}  -(3)^2 +10\

&= underbrace{(x+3)^2}_{(a+b)^2} -9 +10\

&= (x+3)^2+1

end{aligned}

$

end{document}

{} so + gets infix spacing, added strut so the braces are all at same depth, and added openup to increase the line spacing a bit.