# How do you simplify (4a^2+11a+6)/(a^2+4a+4) ÷ (4a^2-5a-6)?

Jun 12, 2016

Factor everything to see what you can eliminate.

#### Explanation:

$4 {a}^{2} + 11 a + 6$

$= 4 {a}^{2} + 8 a + 3 a + 6$

$= 4 a \left(a + 2\right) + 3 \left(a + 2\right)$

$= \left(4 a + 3\right) \left(a + 2\right)$

${a}^{2} + 4 a + 4$

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

$= \left(a + 2\right) \left(a + 2\right)$

$4 {a}^{2} - 5 a - 6$

$= 4 {a}^{2} - 8 a + 3 a - 6$

$= 4 a \left(a - 2\right) + 3 \left(a - 2\right)$

$= \left(4 a + 3\right) \left(a - 2\right)$

Now that everything has been factored, we can put our expression back into rational form.

$\frac{\left(4 a + 3\right) \left(a + 2\right)}{\left(a + 2\right) \left(a + 2\right)} \times \frac{1}{\left(4 a + 3\right) \left(a - 2\right)}$

$= \frac{\cancel{4 a + 3} \cancel{a + 2}}{\cancel{a + 2} \left(a + 2\right)} \times \frac{1}{\cancel{4 a + 3} \left(a - 2\right)}$

$= \frac{1}{\left(a + 2\right) \left(a - 2\right)}$

$= \frac{1}{{a}^{2} - 4}$

Hopefully this helps!