2x^2+7x+3/4x+12 perform all operations and simplify

To simplify the expression \(\frac{2x^2 + 7x + 3}{4x + 12}\), we will follow these steps:

1. Factorize both the numerator and the denominator, if possible.
2. Simplify by canceling common factors.

Let's start by factorizing the denominator.

### Step 1: Factorize the Denominator
The denominator is \(4x + 12\). We can factor out the common factor of 4:
\[ 4x + 12 = 4(x + 3) \]

### Step 2: Factorize the Numerator
The numerator is \(2x^2 + 7x + 3\). We need to factorize this quadratic expression.

To factorize \(2x^2 + 7x + 3\):
- Look for two numbers that multiply to \(2 \cdot 3 = 6\) and add up to 7.
- These numbers are 6 and 1.

Thus, we rewrite the middle term \(7x\) as \(6x + x\):
\[ 2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 \]

Now, group the terms:
\[ 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) \]

Factor out the common binomial factor \((x + 3)\):
\[ 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) \]

### Step 3: Simplify the Expression
Now the expression becomes:
\[ \frac{2x^2 + 7x + 3}{4x + 12} = \frac{(2x + 1)(x + 3)}{4(x + 3)} \]

Cancel the common factor \((x + 3)\):
\[ \frac{(2x + 1)(x + 3)}{4(x + 3)} = \frac{2x + 1}{4} \]

So, the simplified expression is:
\[ \frac{2x + 1}{4} \]

Would you like more details or have any questions?

Here are some questions you might want to ask:
1. How do you factorize a quadratic expression?
2. What if the numerator or denominator cannot be factorized?
3. Can you show more examples of rational expressions?
4. How do you handle complex rational expressions?
5. What is the difference between simplifying and solving an expression?

**Tip:** Always check if the terms in the numerator and denominator can be factorized before attempting to simplify rational expressions. This can help you identify and cancel common factors easily.

Learn how to simplify the rational expression (2x^2 + 7x + 3) / (4x + 12) step-by-step. This problem involves factoring quadratic expressions and simplifying rational expressions. Suitable for high school students studying algebra and rational expressions.

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