To simplify the expression \((x^2 - 5x + 6)^2\), follow these steps:

1. Factorize the quadratic expression inside the square:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

2. Substitute this factorized form back into the original expression:
\[ (x^2 - 5x + 6)^2 = ((x - 2)(x - 3))^2 \]

3. Use the property of exponents \((ab)^2 = a^2b^2\) to expand:
\[ ((x - 2)(x - 3))^2 = (x - 2)^2 (x - 3)^2 \]

Now, expand each binomial square:
\[ (x - 2)^2 = x^2 - 4x + 4 \]
\[ (x - 3)^2 = x^2 - 6x + 9 \]

4. Multiply these expanded forms together:
\[ (x^2 - 4x + 4)(x^2 - 6x + 9) \]

5. Perform the multiplication (using the distributive property):
\[ 
(x^2 - 4x + 4)(x^2 - 6x + 9) = x^2(x^2 - 6x + 9) - 4x(x^2 - 6x + 9) + 4(x^2 - 6x + 9) 
\]

Expand each term separately:
\[ x^2(x^2 - 6x + 9) = x^4 - 6x^3 + 9x^2 \]
\[ -4x(x^2 - 6x + 9) = -4x^3 + 24x^2 - 36x \]
\[ 4(x^2 - 6x + 9) = 4x^2 - 24x + 36 \]

Combine all terms together:
\[ x^4 - 6x^3 + 9x^2 - 4x^3 + 24x^2 - 36x + 4x^2 - 24x + 36 \]

Combine like terms:
\[ x^4 - 10x^3 + 37x^2 - 60x + 36 \]

So, the simplified form of \((x^2 - 5x + 6)^2\) is:
\[ x^4 - 10x^3 + 37x^2 - 60x + 36 \]

Would you like more details or have any questions?

Here are some related questions for further practice:

1. Simplify \((x^2 + 3x + 2)^2\).
2. Factorize and expand \((x^2 - 4x + 4)^2\).
3. Expand and simplify \((2x^2 - 3x + 1)^2\).
4. Simplify the expression \((x^2 - 7x + 12)^2\).
5. Factorize and expand \((x - 1)^2(x - 5)^2\).

**Tip:** When simplifying or expanding expressions, always look for opportunities to factorize first, as it can make the process much easier.

Explore how to simplify the expression (x² - 5x + 6)² with a detailed step-by-step solution. This problem involves algebraic concepts such as quadratic equations and binomial expansion. Suitable for students in Grades 10-12.

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