# How do you simplify (2x^2+11x+5)/(3x^2+17x+10)?

Oct 8, 2015

$\frac{2 x + 1}{3 x + 2}$

#### Explanation:

Factor the numerator $y 1 = 2 {x}^{2} + 11 x + 5 =$ 2(x + p)(x + q)
Use new AC Method to factor trinomials.
Converted trinomial: ${x}^{2} + 11 x + 10.$
Factor pairs of 10 --> (1, 10). This sum is 11 = b. Then $p = \frac{1}{2}$ and $q = \frac{10}{2} = 5$.
$y 1 = 2 \left(x + \frac{1}{2}\right) \left(x + 5\right) = \left(2 x + 1\right) \left(x + 5\right)$

Next, factor the denominator:
$y 2 = 3 {x}^{2} + 17 x + 10.$
Converted trinomial: ${x}^{2} + 17 x + 30.$
Factor pairs of (30) --> (2, 15). This sum is 17 = b. Then $p = \frac{2}{3}$ and $q = \frac{15}{3} = 5.$
$y 2 = 3 \left(x + \frac{2}{3}\right) \left(x + 5\right) = \left(3 x + 2\right) \left(x + 5\right)$.
Finally: $\frac{y 1}{y 2} = \frac{\left(2 x + 1\right) \left(x + 5\right)}{\left(3 x + 2\right) \left(x + 5\right)} = \frac{2 x + 1}{3 x + 2}$