$\displaystyle{{y}}^{{2}}+{4}{y}={y}{\left({y}+{4}\right)}\usepackage{cmbright}\Huge\$ and $\displaystyle{{y}}^{{2}}+{7}{y}+{12}={\left({y}+{4}\right)}{\left({y}+{3}\right)}\usepackage{cmbright}\Huge\$ , so the LCM $\displaystyle={y}{\left({y}+{4}\right)}{\left({y}+{3}\right)}\usepackage{cmbright}\Huge\$ .

$\displaystyle\frac{{{y}+{3}}}{{{{y}}^{{2}}+{4}{y}}}-\frac{{9}}{{{{y}}^{{2}}+{7}{y}+{12}}}=\frac{{{y}+{3}}}{{{y}{\left({y}+{4}\right)}}}-\frac{{9}}{{{\left({y}+{4}\right)}{\left({y}+{3}\right)}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{\left({y}+{3}\right)}{\left({y}+{3}\right)}}}{{{y}{\left({y}+{4}\right)}{\left({y}+{3}\right)}}}-\frac{{{9}{\left({y}\right)}}}{{{\left({y}+{4}\right)}{\left({y}+{3}\right)}{\left({y}\right)}}}=\frac{{{{y}}^{{2}}+{6}{y}+{9}}}{{{{y}}^{{3}}+{7}{{y}}^{{2}}+{12}{y}}}-\frac{{{9}{y}}}{{{{y}}^{{3}}+{7}{{y}}^{{2}}+{12}{y}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{{y}}^{{2}}-{3}{y}+{9}}}{{{{y}}^{{3}}+{7}{{y}}^{{2}}+{12}{y}}}\usepackage{cmbright}\Huge\$