I think your equation solution is correct but you havwe done it the very long way :)
Before you think about what to get rid of you have to think about whether either side can be simplified.
So it is
Can i simplify the sides independently? If so do it.
What do I want to get rid of? How? Do it - to both sides.
Can i simplify the sides independently? If so do it.
What do I want to get rid of? How? Do it - to both sides.
Can i simplify the sides independently? If so do it.
etc etc
This is why I wanted to see if you can simplify terms properly :)
$$\begin{array}{rrlll}\\\\
2(x-3)+5x&=&4(x+2)-5\qquad&(2x-3)=2x-6\;and\;4(x+2)=4x+8\\\\
2x-6+5x&=&4x+8-5\textcolor[rgb]{0,0,1}{\;\;excellent\; start!\;}\qquad&(+5\;on\;both\;sides)\\\\
&&&\textcolor[rgb]{0,0,1}{\mbox{but now, before you start adding and subtracting things you need to simpliey each side seperately}}\\\\
2x+5x-6&=&4x+8-5\\\\
7x-6&=&4x+3&Now\; add\; 6 \;to \;both \;sides\\\\
7x-6+6&=&4x+3+6&Now\; simplify\\\\
7x&=&4x+9&Now\; subtract\;4x\;from\;each\;side\;\\\\
7x-4x&=&4x-4x+9&Now\; simplify\;\\\\
3x&=&9&Now\; divide\; both \;sides\; by\; 3\;\\\\
\frac{3x}{3}&=&\frac{9}{3}&Now\; simplify\;\\\\
x&=&3&\;\;\\
\end{array}$$
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