https://file.aiquickdraw.com/mathbot/file/1727320808670l6e50z7q.jpg

The equation provided in the image is:

\[
y = \log \left( \frac{(2x^2 + x + 3)^2}{(2x + 2)^3} \right)
\]

To simplify or solve this, we can apply logarithmic properties and simplify the expression inside the logarithm. Here's the step-by-step process:

### Step 1: Apply logarithmic property
Using the property \(\log \left( \frac{a}{b} \right) = \log(a) - \log(b)\), we can split the logarithm:

\[
y = \log((2x^2 + x + 3)^2) - \log((2x + 2)^3)
\]

### Step 2: Simplify powers inside the logarithm
Next, we use the logarithmic property \(\log(a^n) = n \log(a)\) to bring down the exponents:

\[
y = 2 \log(2x^2 + x + 3) - 3 \log(2x + 2)
\]

At this point, the equation is simplified to a form involving two logarithmic terms.

Would you like more details on solving or analyzing this further, or do you have any specific questions about this step?

Here are some related questions for you:

1. Would you like to graph the function \( y = \log \left( \frac{(2x^2 + x + 3)^2}{(2x + 2)^3} \right) \)?
2. Do you want to solve for \(x\) in terms of \(y\)?
3. Would you like to explore limits of this function as \( x \to \infty \) or \( x \to -\infty \)?
4. How would the function change if the logarithm were to a different base, such as base 10 or base \(e\)?
5. Are you interested in finding the domain of this logarithmic function?

**Tip**: Always remember to check the domain when dealing with logarithmic functions. The argument inside the log must be strictly positive.

Learn how to simplify the logarithmic expression y = log[((2x^2 + x + 3)^2) / (2x + 2)^3] using logarithmic properties and exponent laws. This problem covers key algebraic concepts suitable for Grades 10-12.

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