Question

3x^2+4x+3 / + 3x^2 + 6x

Which is the correct polynomial

6x^2+24x+3
9x^2+10x+3
6x^2+10x+3
9x^2+24x+3

Answer 1

To simplify the expression \(\frac{3x^2 + 4x + 3}{3x^2 + 6x}\), we can start by checking the factors of the numerator and the denominator:

### Step 1: Factor the denominator
The denominator \(3x^2 + 6x\) can be factored as:
\[
3x(x + 2)
\]

### Step 2: Check the numerator
Next, we leave the numerator \(3x^2 + 4x + 3\) as it is for now since it doesn’t factor neatly.

### Step 3: Division of polynomials
To simplify the fraction:
\[
\frac{3x^2 + 4x + 3}{3x^2 + 6x}
\]

We can perform polynomial long division or synthetic division, but since the degree of the numerator is equal to that of the denominator, we can divide the leading coefficients:

1. The leading term of the numerator is \(3x^2\).
2. The leading term of the denominator is \(3x^2\).

Dividing:
\[
\frac{3x^2}{3x^2} = 1
\]

### Step 4: Subtract and continue
Now, we will subtract \(1 \cdot (3x^2 + 6x)\) from the numerator:
\[
3x^2 + 4x + 3 - (3x^2 + 6x) = 4x + 3 - 6x = -2x + 3
\]

### Step 5: Complete the fraction
So, we have:
\[
\frac{3x^2 + 4x + 3}{3x^2 + 6x} = 1 + \frac{-2x + 3}{3x^2 + 6x}
\]

However, to solve the expression given, we need to work at the level of original polynomials given to determine what choices correlate.

### Identifying Correct Polynomial
Given that we’re comparing to options, let's multiply it out:
1. If we directly look at:
\(\frac{3x^2 + 4x + 3}{3x^2 + 6x}\), we proceed by evaluating values for \(x\).

To obtain a clear answer directly from the options without the need for extensive expansion:
None of the provided options seem to result from any clear relationship from \(3x^2 + 4x + 3\) divided or related cleanly back to particular options of `6x^2 + 24x + 3`, `9x^2 + 10x + 3`, `6x^2 + 10x + 3`, or `9x^2 + 24x + 3`.

### Conclusion
Since this isn't quite simple to derive without any additional context, I would suggest simplifying further or re-evaluating potential values of x to confirm which polynomial representation simplifies equivalently. If additional context indicates certain values, we could specify more directly.

To answer **which polynomial is correct** from the given choices, please review calculated relations, as none directly correlate without further substitution or specific evaluations to assure alignment with given polynomials.