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We have to determine the values for which the rational expression is undefined:

$鈸�\frac{3y}{x}鈸�\frac{8n鈭�5}{3n+1}鈸�\frac{a+10}{a^2+4a+3}$

The expression will be undefined when the denominator is zero.

Set the denominator equal to zero. Solve for the variable.

$x=0$

$\mathrm{The}\mathrm{rational}\mathrm{expression}\frac{3y}{x}\mathrm{is}\mathrm{undefined}\mathrm{for}x=0.$

The expression will be undefined when the denominator is zero.

Set the denominator equal to zero. Solve for the variable.

$$\begin{gathered} 3n+1=0 \\ 3n+1-1=0-1 \\ 3n=-1 \\ \frac{3}{3}n=-\frac{1}{3} \\ n=-\frac{1}{3} \end{gathered}$$

$\mathrm{The}\mathrm{rational}\mathrm{expression}\frac{8n鈭�5}{3n+1}\mathrm{is}\mathrm{undefined}\mathrm{for}n=-\frac{1}{3}.$

The expression will be undefined when the denominator is zero.

Set the denominator equal to zero. Solve for the variable.

$$\begin{gathered} a^2+4a+3=0 \\ {a}^2+(3+1)a+3=0 \\ {a}^2+3a+a+3=0 \\ a(a+3)+1(a+3)=0 \\ (a+1)(a+3)=0 \\ a+1=0\mathrm{or}\mathrm{a}+3=0 \\ a=-1\mathrm{or}\mathrm{a}=-3 \end{gathered}$$

$\mathrm{The}\mathrm{rational}\mathrm{expression}\frac{a+10}{a^2+4a+3}\mathrm{is}\mathrm{undefined}\mathrm{for}\mathrm{a}=-1\mathrm{and}\mathrm{a}=-3.$