https://r2.erweima.ai/mathbot/image/1725897334026dg3382m2.jpg

The equation in the image is:

\[
(x + 1)^{x^2 + 7x + 10} = (2x + 3)^{x^2 + 7x + 10}.
\]

To solve this equation, let's analyze it step by step:

### Step 1: Simplify the Exponents
Since the exponents on both sides of the equation are the same, we can equate the bases:

\[
x + 1 = 2x + 3.
\]

### Step 2: Solve for \(x\)
Now, solve the linear equation:

\[
x + 1 = 2x + 3.
\]

Subtract \(x\) from both sides:

\[
1 = x + 3.
\]

Subtract 3 from both sides:

\[
x = -2.
\]

### Step 3: Check for Validity
Substituting \(x = -2\) into the original bases:

- For the left side: \(x + 1 = -2 + 1 = -1\).
- For the right side: \(2x + 3 = 2(-2) + 3 = -4 + 3 = -1\).

Since both bases become \(-1\), and the exponents are equal, the equation holds true.

### Final Solution
The solution is:

\[
\boxed{x = -2.}
\]

Would you like further details or have any questions?

### Related Questions:
1. How can equations with exponential terms be simplified by comparing their bases?
2. What happens if both sides of the equation have negative bases with even exponents?
3. How would the solution change if the exponents were not the same on both sides?
4. What other methods could be used to solve exponential equations?
5. How can graphing be used to verify the solution of exponential equations?

**Tip:** When exponential terms have the same exponents, equating the bases can simplify the problem significantly!

(x + 1)^(x^2 + 7x + 10) = (2x + 3)^(x^2 + 7x + 10).

This page provides a step-by-step solution for solving the exponential equation (x + 1)^(x^2 + 7x + 10) = (2x + 3)^(x^2 + 7x + 10). Key algebraic concepts, such as comparing exponential bases, are explored. Suitable for students in Grades 10-12.

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