Perform the following operation and express in simplest form. start fraction, x, squared, minus, 4, divided by, 4, x, plus, 12, end fraction, divided by, start fraction, x, squared, plus, 5, x, plus, 6, divided by, x, squared, plus, 3, x, end fraction 4x+12 x 2 −4 ​ ÷ x 2 +3x x 2 +5x+6 ​

Answer 1

To simplify this expression, we first factor all the expressions:
1. x^2 - 4 can be factored into (x+2)(x-2)
2. x^2 + 5x + 6 can be factored into (x+3)(x+2)
3. x^2 + 3x can be factored into x(x+3)

Now, the given expression becomes:
((x+2)(x-2)/(4x+12)) ÷ ((x+3)(x+2)/(x(x+3)))

Next, we simplify this expression by multiplying by the reciprocal of the divisor and simplifying:
((x+2)(x-2)/(4(x+3))) * (x(x+3)/((x+3)(x+2)))
= (x^2-4)/(4(x+3)) * (x(x+3)/(x+3)(x+2))
= (x+2)(x-2)*x/(4*(x+3)(x+2))
= x(x-2)/4(x+3)
= (x^2-2x)/4(x+3)

Therefore, the simplified form of the given expression is (x^2-2x)/4(x+3).