Factoring a Quadratic Equation:
To solve the quadratic equation 9x² - 9(a+b)x + (2a² + 5ab + 2b²) = 0, we need to factorize it. The equation is in the form ax² + bx + c = 0.

Step 1: Identify Coefficients
In this equation, a = 9, b = -9(a+b), and c = 2a² + 5ab + 2b².

Step 2: Factorize
To factorize the equation, we need to find two numbers that multiply to a*c (18b²) and add up to b (-9(a+b)).

Step 3: Find the Factors
The factors of 18b² that add up to -9(a+b) are -6b and -3b.

Step 4: Rewrite the Equation
Now we rewrite the middle term -9(a+b)x as -6bx - 3bx.
9x² - 6bx - 3bx + 2a² + 5ab + 2b² = 0

Step 5: Grouping
Group the terms and factorize by grouping.
(9x² - 6bx) - (3bx - 2a²) + 5ab + 2b² = 0
3x(3x - 2b) - b(3x - 2a) + 5ab + 2b² = 0

Step 6: Factorize Completely
Now factorize the grouped terms.
(3x - b)(3x - 2a) + 5ab + 2b² = 0

Final Answer:
The factored form of the quadratic equation 9x² - 9(a+b)x + (2a² + 5ab + 2b²) = 0 is (3x - b)(3x - 2a) + 5ab + 2b² = 0.