To solve the equation, we first need to express it correctly:

Given equation:

$32(4)^{2x} = 8^3 \times (4^{x+1})$

Step 1: Simplify the equation

We know that: $32 = 2^5 \quad \text{and} \quad 4 = 2^2 \quad \text{and} \quad 8 = 2^3$

So, let's rewrite the equation using base 2: $2^5 \times (2^2)^{2x} = (2^3)^3 \times (2^2)^{x+1}$

Simplify the exponents: $2^5 \times 2^{4x} = 2^9 \times 2^{2(x+1)}$

$2^{5+4x} = 2^{9 + 2x + 2}$

Simplify further: $2^{4x+5} = 2^{2x+11}$

Since the bases are the same, we can set the exponents equal to each other: $4x + 5 = 2x + 11$

Step 2: Solve for $x$

Subtract $2x$ from both sides: $2x + 5 = 11$

Subtract 5 from both sides: $2x = 6$

Divide by 2: $x = 3$

So, the solution is $x = 3$.

Do you want details or have any questions? Here are 8 related questions:

1. How can you solve exponential equations using logarithms?
2. What are the properties of exponents that can be used to simplify equations?
3. How would the solution change if the exponents were different?
4. What is the general method for solving exponential equations?
5. How do you convert logarithmic equations to exponential form?
6. How can you solve an equation involving multiple bases?
7. What is the significance of the base in an exponential function?
8. How can graphing help to solve exponential equations?

Tip: Always check your final answer by substituting it back into the original equation.