Perform the following operation and express in simplest form. start fraction, x, squared, minus, 4, divided by, 5, x, plus, 30, end fraction, divided by, start fraction, x, squared, plus, 11, x, plus, 18, divided by, 3, x, plus, 27, end fraction 5x+30 x 2 −4 ​ ÷ 3x+27 x 2 +11x+18 ​

To simplify this expression, start by factoring the numerator and denominator of each fraction.

Numerator of the first fraction, x^2 - 4:
x^2 - 4 = (x - 2)(x + 2)

Denominator of the first fraction, 5x + 30:
5x + 30 = 5(x + 6)

Numerator of the second fraction, x^2 + 11x + 18:
x^2 + 11x + 18 = (x + 2)(x + 9)

Denominator of the second fraction, 3x + 27:
3x + 27 = 3(x + 9)

Now substitute the factored forms back into the expression:

[(x - 2)(x + 2) / 5(x + 6)] / [(x + 2)(x + 9) / 3(x + 9)]

When dividing fractions, you flip the second fraction and multiply:

[(x - 2)(x + 2) / 5(x + 6)] * [3(x + 9) / (x + 2)(x + 9)]

Now, cancel out common factors between numerators and denominators:

= [(x - 2) * 3] / [5 * (x + 6)]

= (3x - 6) / (5x + 30)

= 3(x - 2) / 5(x + 6)

Therefore, the expression simplifies to 3(x - 2) / 5(x + 6)