Answer:
(a + 1)²(a + 7).
Step-by-step explanation:
The original expression:
[tex]\frac{a^{3}}{a^{2}+2a+1} -\frac{3}{a^{2}+8a+7}[/tex]
Factor each denominator:
[tex]= \frac{a^{3}}{(a+1)(a+1)} - \frac{3}{(a+1)(a+7)}[/tex]
Factor out [tex]\frac{ 1}{a+1 }[/tex]
[tex]= \frac{ 1}{a+1 }(\frac{a^{3}}{a+1} - \frac{3}{a+7})[/tex]
Multiply each term by the opposite denominator and add:
[tex]= \frac{ 1}{a+1 }[\frac{a^{3}(a+7)-3(a+1)}{(a+1)(a+7)}][/tex]
Remove brackets:
[tex]= \frac{a^{3}(a+7)-3(a+1)}{(a+1)^{2}(a+7)}[/tex]
The least common denominator is (a + 1)²(a + 7).
