Simplify the rational expression.
2x/x^2+4x+3 divided by 1/x+3 plus 2/x+1
To simplify the rational expression, we need to find a common denominator for the fractions in the numerator and denominator.
Let's start by simplifying the expression in the numerator. We have 2x divided by (x^2 + 4x + 3).
The numerator, 2x, is already simplified. To simplify the denominator, we can factorize it: (x^2 + 4x + 3) = (x + 1)(x + 3).
Now, let's move on to the expression in the denominator: 1/(x + 3) + 2/(x + 1).
The first term, 1/(x + 3), is already simplified. The second term, 2/(x + 1), can be rewritten with a common denominator of (x + 3): 2(x + 3)/(x + 1)(x + 3) = 2x + 6/(x + 1)(x + 3).
Now, we can substitute these simplified expressions into the original expression:
2x/(x^2 + 4x + 3) ÷ (1/(x + 3) + 2/(x + 1))
= 2x/[(x + 1)(x + 3)] ÷ [1/(x + 3) + (2x + 6)/(x + 1)(x + 3)]
Now, let's simplify further by multiplying the numerator and denominator of the main fraction by the common denominator, (x + 1)(x + 3):
2x/[(x + 1)(x + 3)] ÷ [1/(x + 3) + (2x + 6)/(x + 1)(x + 3)]
= [2x(x + 1)(x + 3)] / [(x + 3)(x + 1)(x + 3)] ÷ [1(x + 1)(x + 3) + (2x + 6)] / [(x + 1)(x + 3)]
= [2x(x + 1)(x + 3)] / [(x + 1)(x + 3)(x + 3)] ÷ [(x + 1)(x + 3) + (2x + 6)] / [(x + 1)(x + 3)]
Next, we can simplify and cancel out common factors:
= [2x(x + 1)(x + 3)] / [(x + 1)(x + 3)(x + 3)] ÷ [2(x + 1)(x + 3)] / [(x + 1)(x + 3)]
= [2x(x + 1)(x + 3)] / [(x + 1)(x + 3)(x + 3)] * [(x + 1)(x + 3)] / [2(x + 1)(x + 3)]
= [2x(x + 1)(x + 3)] / [(x + 1)(x + 3)(x + 3)] * [(x + 1)(x + 3)] / [2(x + 1)(x + 3)]
Notice that all the (x + 1)(x + 3) terms cancel out, leaving us with:
= 2x / (x + 3)
Therefore, the simplified expression is 2x / (x + 3).