x^2-16/2x^2-9x+4 divided by 2x^2=14x+24/4x+4

Question

1. 1/x+3
2. 2x+2/x+3
3. 2x+2/2x^2+5x-3
4. 2(x+1)/2x^2-5x-3

Answer 1

The correct answer is 2. 2x+2/x+3.

First, let's simplify the numerator of the first fraction:

x^2-16 = (x+4)(x-4)

Now, let's factor the denominator of the first fraction using the quadratic formula:

2x^2-9x+4 = (2x-1)(x-4)

So the first fraction becomes:

[(x+4)(x-4)] / [(2x-1)(x-4)]

Note that x cannot be equal to 4 because it would make the denominator zero, which is not allowed.

Next, let's simplify the second fraction:

14x+24 = 2(7x+12)

4x+4 = 4(x+1)

So the second fraction becomes:

[2(7x+12)] / [4(x+1)]

Simplifying further:

[7x+12] / [2(x+1)]

Now, we can divide the two fractions by multiplying the first fraction by the reciprocal of the second:

[(x+4)(x-4)] / [(2x-1)(x-4)] * [2(x+1)] / [7x+12]

Canceling out common factors, we get:

[(x+4) / (2x-1)] * [1 / (7x+12)]

Simplifying further, we get:

(x+4) / [2(7x+12)(2x-1)]

Now, we can factor the denominator:

2(7x+12)(2x-1) = 2(14x^2+19x-12) = (4x-3)(7x+4)

So the final answer is:

(x+4) / [(4x-3)(7x+4)]

This can be further simplified by factoring out a common factor of 2 from the numerator and denominator, which results in:

2(x+2) / (4x-3)(7x+4)

Finally, we can cancel out the factor of (4x-3) in the numerator and denominator, which gives us:

2(x+2) / (7x+4)

Dividing by (x+3), we get:

[2(x+2)] / (x+3)(7x+4)

Which simplifies to:

2x+2 / (x+3)