# How do you find the quotient of (a^3+4a^2+7a+6) /( a+2)?

Nov 9, 2015

Start with dividing the term with the highest exponent in the numerator by the term with the highest exponent in the denominator, so here, divide ${a}^{3}$ by $a$:
${a}^{3} / a = {a}^{2}$.

So, your first term needs to be ${a}^{2} \left(a + 2\right)$.
$\implies$ Transform the numerator into ${a}^{2} \left(a + 2\right) + 2 {a}^{2} + 7 a + 6$.

Always pay attention that the value of the numerator doesn't change - here, you have "splitted" the term $4 {a}^{2}$ into the part in ${a}^{2} \left(a + 2\right)$ and the rest: $2 {a}^{2}$.

So far, you've got:
$\frac{{a}^{3} + 4 {a}^{2} + 7 a + 6}{a + 2}$
$= \frac{{a}^{2} \left(a + 2\right) + 2 {a}^{2} + 7 a + 6}{a + 2}$
$= \frac{{a}^{2} \left(a + 2\right)}{a + 2} + \frac{2 {a}^{2} + 7 a + 6}{a + 2}$
$= {a}^{2} + \frac{2 {a}^{2} + 7 a + 6}{a + 2}$

Your fraction is smaller now. Proceed in the same way:
- divide $2 {a}^{2}$ by $a$, result: $\frac{2 {a}^{2}}{a} = 2 a$
- create the expression $2 a \left(a + 2\right)$ in the numerator.
- take $4 a$ (part of your new expression) from $7 a$

Now you have:
${a}^{2} + \frac{2 {a}^{2} + 7 a + 6}{a + 2}$
$= {a}^{2} + \frac{2 a \left(a + 2\right) + 3 a + 6}{a + 2}$
$= {a}^{2} + \frac{2 a \left(a + 2\right)}{a + 2} + \frac{3 a + 6}{a + 2}$
$= {a}^{2} + 2 a + \frac{3 a + 6}{a + 2}$

The last part is easy: $3 a$ divided by $a$ is $3$, and the last expression can be factorized cleanly in $3 \left(a + 2\right) = 3 a + 6$.

${a}^{2} + 2 a + \frac{3 a + 6}{a + 2}$
$= {a}^{2} + 2 a + \frac{3 \left(a + 2\right)}{a + 2}$
$= {a}^{2} + 2 a + 3$

$\frac{{a}^{3} + 4 {a}^{2} + 7 a + 6}{a + 2} = {a}^{2} + 2 a + 3$