Create a question about finding roots for a challenging quadratic equation by expanding and then factoring. The equation has brackets on both sides. It can have a constant or a bracket can be a perfect square. Show the solution. The solution must be factorable.

### Question:

Solve for \( x \) in the following quadratic equation:

\[
(2x - 3)(x + 4) = (x - 1)^2
\]

### Solution:

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

Expand the left-hand side:

\[
(2x - 3)(x + 4) = 2x(x + 4) - 3(x + 4)
\]
\[
= 2x^2 + 8x - 3x - 12
\]
\[
= 2x^2 + 5x - 12
\]

Expand the right-hand side:

\[
(x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1
\]

Now the equation becomes:

\[
2x^2 + 5x - 12 = x^2 - 2x + 1
\]

#### Step 2: Move all terms to one side

Subtract \( x^2 - 2x + 1 \) from both sides:

\[
2x^2 + 5x - 12 - (x^2 - 2x + 1) = 0
\]
\[
2x^2 + 5x - 12 - x^2 + 2x - 1 = 0
\]
\[
x^2 + 7x - 13 = 0
\]

#### Step 3: Factor the quadratic equation

We now need to factor \( x^2 + 7x - 13 = 0 \). This quadratic is factorable as follows:

Find two numbers that multiply to give \( -13 \) and add to give \( 7 \). These numbers are \( 13 \) and \( -1 \).

Thus, we factor the equation as:

\[
(x + 13)(x - 1) = 0
\]

#### Step 4: Solve for \( x \)

Set each factor equal to zero:

\[
x + 13 = 0 \quad \text{or} \quad x - 1 = 0
\]

Thus, the solutions are:

\[
x = -13 \quad \text{or} \quad x = 1
\]

### Final Answer:

The roots of the equation are \( x = -13 \) and \( x = 1 \).

---

Would you like further details or have any questions?

Here are 5 additional problems related to this topic:
1. Solve \( (3x + 2)(x - 5) = (x + 1)^2 \).
2. Solve \( (x - 6)(2x + 1) = (x - 2)(x + 5) \).
3. Factor and solve the quadratic equation \( 3x^2 + 7x - 6 = 0 \).
4. Expand and solve \( (2x - 1)(x + 3) = x^2 + 2x - 4 \).
5. Find the roots of \( (x - 4)^2 = (2x + 1)(x - 2) \).

**Tip**: When expanding quadratic equations with brackets, always combine like terms carefully before factoring to avoid errors!

Learn how to solve challenging quadratic equations with brackets on both sides by expanding and then factoring. Follow a detailed step-by-step solution to find the roots of the equation (2x - 3)(x + 4) = (x - 1)^2, suitable for students in Grades 9-10.

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