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To solve this problem, we need to divide the polynomial \(6m^3 + 14m^2 - 2m + 3\) by the binomial \(2m + 4\).

### Step 1: Factor the Denominator
The denominator \(2m + 4\) can be factored:
\[
2m + 4 = 2(m + 2).
\]

### Step 2: Polynomial Long Division
We will perform polynomial long division of \(6m^3 + 14m^2 - 2m + 3\) by \(2(m + 2)\).

1. Divide the leading term of the numerator \(6m^3\) by the leading term of the denominator \(2m\):
   \[
   \frac{6m^3}{2m} = 3m^2.
   \]

2. Multiply \(3m^2\) by the entire divisor \(2(m + 2)\):
   \[
   3m^2 \times 2(m + 2) = 6m^3 + 12m^2.
   \]

3. Subtract this result from the original polynomial:
   \[
   (6m^3 + 14m^2 - 2m + 3) - (6m^3 + 12m^2) = 2m^2 - 2m + 3.
   \]

4. Divide the new leading term \(2m^2\) by \(2m\):
   \[
   \frac{2m^2}{2m} = m.
   \]

5. Multiply \(m\) by the entire divisor:
   \[
   m \times 2(m + 2) = 2m^2 + 4m.
   \]

6. Subtract this result:
   \[
   (2m^2 - 2m + 3) - (2m^2 + 4m) = -6m + 3.
   \]

7. Divide the new leading term \(-6m\) by \(2m\):
   \[
   \frac{-6m}{2m} = -3.
   \]

8. Multiply \(-3\) by the divisor:
   \[
   -3 \times 2(m + 2) = -6m - 6.
   \]

9. Subtract this result:
   \[
   (-6m + 3) - (-6m - 6) = 9.
   \]

### Step 3: Write the Final Quotient and Remainder
The quotient is \(3m^2 + m - 3\) and the remainder is \(9\). Therefore, the division can be expressed as:
\[
\frac{6m^3 + 14m^2 - 2m + 3}{2m + 4} = 3m^2 + m - 3 + \frac{9}{2(m + 2)}.
\]

Since we are asked to simplify the answer, the final simplified expression is:
\[
3m^2 + m - 3 + \frac{9}{2(m + 2)}.
\]

### Final Answer:
\[
\frac{6m^3 + 14m^2 - 2m + 3}{2m + 4} = 3m^2 + m - 3 + \frac{9}{2(m + 2)}.
\]

Would you like a further breakdown of any steps or have any questions?

### Related Questions:
1. How do you perform polynomial long division for other polynomials?
2. What are the conditions for a polynomial to be exactly divisible by another polynomial?
3. How can polynomial division be used to simplify complex rational expressions?
4. What are the similarities and differences between polynomial division and numerical division?
5. How does synthetic division differ from long division, and when is it used?

### Tip:
Always factor the divisor completely to check for possible simplifications before performing polynomial long division. This can sometimes make the division process easier or reveal simpler forms of the quotient.

Perform the division: (6m^3 + 14m^2 - 2m + 3) / (2m + 4)

This page explains how to perform polynomial long division on (6m^3 + 14m^2 - 2m + 3) divided by (2m + 4), including factoring and step-by-step calculations. Suitable for high school students learning advanced algebra concepts.

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