We are given the function $g(x) = x^2 - 4x$, and we need to solve the equation $g(x) = 45$.

This translates to:

$x^2 - 4x = 45$

Step 1: Rearrange the equation

Move all terms to one side to set the equation equal to zero:

$x^2 - 4x - 45 = 0$

Step 2: Solve the quadratic equation

We will solve the quadratic equation $x^2 - 4x - 45 = 0$ using factoring. First, we look for two numbers that multiply to $-45$ and add up to $-4$.

The two numbers are $-9$ and $5$ because:

$-9 \times 5 = -45 \quad \text{and} \quad -9 + 5 = -4$

Thus, we can factor the quadratic as:

$(x - 9)(x + 5) = 0$

Step 3: Solve for $x$

Now, we set each factor equal to zero:

$x - 9 = 0 \quad \Rightarrow \quad x = 9$ $x + 5 = 0 \quad \Rightarrow \quad x = -5$

Final Answer:

The solutions to the equation $g(x) = 45$ are:

$x = 9 \quad \text{and} \quad x = -5$

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Would you like further clarification or details on any steps? Here are five related questions for you to explore:

1. How do you solve a quadratic equation using the quadratic formula?
2. Can you factor quadratic equations that don't have integer solutions?
3. What is the discriminant, and how does it relate to the number of solutions of a quadratic equation?
4. How do you solve a quadratic equation by completing the square?
5. How would you graph the function $g(x) = x^2 - 4x$?

Tip: Always check the factorization of a quadratic by expanding the factored form to ensure it's correct before solving.