Let's solve the equation \((x - 3)(x - 4) = (x + 5)^2\).

### Step 1: Expand both sides of the equation

**Left-hand side:**
\[
(x - 3)(x - 4) = x^2 - 4x - 3x + 12 = x^2 - 7x + 12
\]

**Right-hand side:**
\[
(x + 5)^2 = (x + 5)(x + 5) = x^2 + 10x + 25
\]

### Step 2: Set up the equation
Now substitute the expanded forms back into the original equation:
\[
x^2 - 7x + 12 = x^2 + 10x + 25
\]

### Step 3: Simplify the equation
Subtract \(x^2\) from both sides:
\[
-7x + 12 = 10x + 25
\]

### Step 4: Combine like terms
Move all terms involving \(x\) to one side and constants to the other:
\[
-7x - 10x = 25 - 12
\]
\[
-17x = 13
\]

### Step 5: Solve for \(x\)
\[
x = \frac{13}{-17} = -\frac{13}{17}
\]

### Final Solution:
The solution to the equation is \(x = -\frac{13}{17}\).

Would you like more details or explanations?

Here are 5 related questions:
1. How would you solve the equation if it involved cubed terms?
2. Can you factor quadratic expressions with different coefficients?
3. How do you solve equations that involve multiple variables?
4. How does completing the square help in solving quadratic equations?
5. What is the significance of the discriminant in solving quadratic equations?

**Tip:** Always check for potential extraneous solutions when solving equations with squaring.

Learn how to solve the quadratic equation (x - 3)(x - 4) = (x + 5)^2 step-by-step. This problem focuses on expanding binomials, simplifying equations, and solving for x. Ideal for students in Grades 9-10 learning algebraic equation solving.

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