https://file.aiquickdraw.com/mathbot/file/17264946647847pw1qdpy.jpg

The equation in the image is:

\[
25x^2 + 2x + 2 = 125^{x-2}
\]

Let's solve this step by step.

### Step 1: Rewrite the equation
First, we want to simplify both sides. Notice that \( 25 = 5^2 \) and \( 125 = 5^3 \), so we can express the powers of 25 and 125 in terms of base 5:

\[
(5^2)x^2 + 2x + 2 = (5^3)^{x-2}
\]

Simplifying the right-hand side:

\[
5^{3(x-2)} = 5^{3x - 6}
\]

Thus, the equation becomes:

\[
5^{2x^2} + 2x + 2 = 5^{3x - 6}
\]

### Step 2: Analyze powers
We now have powers of 5 on both sides of the equation. For this to hold true, we will generally need the exponents on both sides to match. However, the equation is not in a form that allows for direct comparison of exponents. At this stage, it's advisable to attempt solving this equation numerically or graphically.

Let me solve it numerically.It seems that the equation cannot be solved analytically with standard methods due to the combination of exponential and polynomial terms. Therefore, it is best to solve this equation using numerical or graphical methods.

Let me attempt a numerical solution using an approximation method.It seems like I can’t do more advanced data analysis right now. Please try again later.

Meanwhile, you can approach solving the equation by using a graphing calculator or online numerical solvers to estimate the value of \( x \). 

Would you like help with any of the following related topics?
1. Solving exponential equations using logarithms.
2. How to use graphing tools for solving equations.
3. Understanding numerical methods for solving equations.
4. Simplifying complex equations by approximation techniques.
5. How to solve equations involving different bases.

**Tip:** For mixed polynomial-exponential equations, numerical methods like Newton's method or using a graphing calculator often provide quicker insights than purely algebraic methods.

Learn how to solve the complex equation involving both exponential and polynomial terms: 25x^2 + 2x + 2 = 125^(x-2). This problem addresses key algebraic concepts like exponentiation rules and polynomial solutions, suitable for students in Grades 10-12.

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