To find the solution to the equation, we need to consider the cases where the absolute value function changes sign.

Case 1: [tex]x+3 \geq 0[/tex]
In this case, the absolute value function becomes [tex]|x+3| = x+3[/tex].

Substitute into the original equation:
[tex]x^{2} +3x+x+3 =0[/tex]
[tex]x^{2} +4x+3 =0[/tex]
tex(x+3) =0[/tex]

So, [tex]x = -1[/tex] or [tex]x=-3[/tex].

Case 2: [tex]x+3 < 0[/tex]
In this case, the absolute value function becomes [tex]|x+3| = -(x+3) = -x-3[/tex].

Substitute into the original equation:
[tex]x^{2} +3x-x-3 =0[/tex]
[tex]x^{2} +2x-3 =0[/tex]
tex(x-1) =0[/tex]

So, [tex]x = -3[/tex] or [tex]x=1[/tex].

Therefore, the solutions to the equation [tex]x^{2} +3x+|x+3|=0[/tex] are [tex]x=-1, -3, 1[/tex].