Answer:
[tex]\dfrac{5}{y+5}[/tex]
Step-by-step explanation:
Here, we have to find the sum of 2 fractions:
1st fraction: [tex]\dfrac{3y}{y^{2}+7y+10}[/tex]
2nd fraction: [tex]\dfrac{2}{y+2}[/tex]
Considering the denominator of 1st fraction:
[tex]y^{2}+7y+10[/tex]
Using factorization method:
[tex]7y[/tex] can be written as [tex](2y + 5y)[/tex].
[tex]\Rightarrow y^{2}+2y+5y+10[/tex]
Taking 5 common from [tex]5y+10[/tex] and y common from [tex]y^{2}+2y[/tex]: [tex]\Rightarrow y(y+2)+5(y+2)[/tex]
Now taking [tex](y+2)[/tex] common:
[tex]\Rightarrow (y+5)(y+2)[/tex]
[tex]\dfrac{3y}{y^{2}+7y+10}[/tex] can be written as [tex]\dfrac{3y}{(y+5)(y+2)}[/tex]
Now, calculating the sum:
[tex]\dfrac{2y}{(y+5)(y+2)} + \dfrac{2}{y+2}[/tex]
Taking LCM and solving:
[tex]\Rightarrow \dfrac{3y+2(y+5)}{(y+5)(y+2)}\\\Rightarrow \dfrac{5y+10}{(y+5)(y+2)}\\\Rightarrow \dfrac{5(y+2)}{(y+5)(y+2)}\\\Rightarrow \dfrac{5}{(y+5)}[/tex]
Hence, answer is [tex]\dfrac{5}{y+5}[/tex].
