3x+6 over 4x+12 divided by x^2-4 over x+3

3x+6 over 4x+12 divided by x^2-4 over x+3

= 3(x+2)/(4(x+3) ) ÷ [(x+2)(x-2)/(x+3)]
= 3(x+2)/(4(x+3)) [ (x+3)/(x+2)(x-2) ]
= 3/(4(x-2) ) , x ≠ +3, ± 2

Right triangle - find length of the missing side. A=16 is the base; B=63 is the side. What is the length of C?

To simplify the expression (3x+6)/(4x+12) ÷ (x^2-4)/(x+3), we can follow these steps:

Step 1: Simplify each numerator and denominator separately within the division.
- In the numerator (3x+6), we can factor out the common factor 3: 3(x+2).
- In the denominator (4x+12), we can factor out the common factor 4: 4(x+3).

After performing step 1, the expression becomes (3(x+2))/(4(x+3)) ÷ (x^2-4)/(x+3).

Step 2: Invert the second fraction (x^2-4)/(x+3) and change the division operation to multiplication.
- The second fraction can be rewritten as (x+3)/(x^2-4). We achieved this by swapping the numerator and denominator of that fraction.

The expression now looks like (3(x+2))/(4(x+3)) * (x+3)/(x^2-4).

Step 3: Multiply the numerators together and the denominators together.
- Multiply the numerators: 3(x+2) * (x+3) = 3(x+2)(x+3).
- Multiply the denominators: 4(x+3) * (x^2-4) = 4(x+3)(x^2-4).

The expression becomes (3(x+2)(x+3))/(4(x+3)(x^2-4)).

Step 4: Simplify further by canceling common factors.
- Notice that both the numerator and denominator have the factor (x+3). Canceling this factor eliminates it from both.

The expression simplifies to 3(x+2)/(4(x^2-4)).

Step 5: Further factorize the denominator.
- The denominator (x^2-4) is a difference of two squares and can be factored as (x+2)(x-2).

The final simplified expression is 3(x+2)/(4(x+2)(x-2)).

This is the simplified form of (3x+6)/(4x+12) ÷ (x^2-4)/(x+3).