Sure! Here's an example of an equation that contains both a polynomial and an exponential function, and the solution involves finding a common denominator:

Equation:

$\frac{3x^2 - 5}{x + 2} = 2^{x+1}$

Steps to solve for $x$:

1. Rewrite the exponential: First, we can simplify the exponential expression on the right-hand side. $2^{x+1} = 2 \cdot 2^x$ So the equation becomes: $\frac{3x^2 - 5}{x + 2} = 2 \cdot 2^x$

2. Isolate the fraction: We now want to eliminate the denominator on the left side. Multiply both sides by $x + 2$: $3x^2 - 5 = (x + 2) \cdot 2 \cdot 2^x$ Simplify: $3x^2 - 5 = 2(x + 2) \cdot 2^x$

3. Distribute: Next, expand the right side: $3x^2 - 5 = 2(x \cdot 2^x + 2 \cdot 2^x)$ Simplifying further: $3x^2 - 5 = 2x \cdot 2^x + 4 \cdot 2^x$

4. Rearrange terms: Move everything to one side to set the equation equal to 0: $3x^2 - 5 - 2x \cdot 2^x - 4 \cdot 2^x = 0$ We can combine like terms to get: $3x^2 - 5 - 2^x(2x + 4) = 0$

5. Solve for $x$: At this point, solving this equation directly is non-trivial and may require numerical methods or further manipulation depending on the context. You might attempt a trial-and-error method or use a graphing calculator, or employ a numerical solver (like Newton's method). Let's try some simple values of $x$:

   For $x = 1$: $3(1)^2 - 5 - 2^1(2(1) + 4) = 3 - 5 - 2 \cdot 6 = -2 - 12 = -14$ So $x = 1$ does not satisfy the equation.

   For $x = 2$: $3(2)^2 - 5 - 2^2(2(2) + 4) = 12 - 5 - 4 \cdot 8 = 7 - 32 = -25$ So $x = 2$ does not satisfy the equation either.

   You would repeat this process or resort to a numerical method (like a graphing calculator) to find the solution. This equation is not easily solved algebraically without tools, but that gives you a general roadmap for tackling this type of problem.

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Do you need further details or clarifications on any step?

Other relevant questions:

1. How can we use numerical methods to solve equations with both polynomials and exponentials?
2. What is Newton's method, and how can it help in finding roots of equations?
3. How would this solution change if the exponential term had a different base, say 3 or $e$?
4. Can a graphing calculator help solve this equation more efficiently?
5. How do we simplify similar equations involving higher-degree polynomials and more complex exponential functions?

Tip: When solving equations involving both exponentials and polynomials, using a graphing approach or numerical solving methods (like Newton's method) can often yield practical solutions, especially when algebraic manipulation is cumbersome.