Math problem-solving tutorial featuring quadratic equations on a black background.

The image contains three quadratic equations. Since no specific one was selected for solving, I will solve all three for completeness.

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Equation 1: x² + 12x + 32 = 0

Step 1: Look for two numbers that multiply to 32 and add to 12.
Those numbers are 8 and 4.

Step 2: Factor the equation:
(x + 8)(x + 4) = 0

Step 3: Set each factor equal to zero:
x + 8 = 0 → x = -8
x + 4 = 0 → x = -4

✔ Solutions: x = -8, x = -4

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Equation 2 (boxed): -x² - 4x + 9 = 0

Step 1: Multiply both sides by -1 to make the x² term positive (easier to work with):
x² + 4x - 9 = 0

Step 2: This doesn’t factor nicely, so use the quadratic formula:
x = [-b ± √(b² - 4ac)] / (2a)
Here, a = 1, b = 4, c = -9

Step 3: Plug in values:
Discriminant = b² - 4ac = 16 - 4(1)(-9) = 16 + 36 = 52
√52 = √(4×13) = 2√13

Step 4: Calculate roots:
x = [-4 ± 2√13] / 2 = -2 ± √13

✔ Solutions: x = -2 + √13, x = -2 - √13

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Equation 3: 3x² - 8x + 5 = 0

Step 1: Try factoring. We need two numbers that multiply to 3×5=15 and add to -8.
Those numbers are -3 and -5.

Step 2: Rewrite middle term:
3x² - 3x - 5x + 5 = 0

Step 3: Group and factor:
(3x² - 3x) - (5x - 5) = 0
3x(x - 1) -5(x - 1) = 0
(3x - 5)(x - 1) = 0

Step 4: Set each factor to zero:
3x - 5 = 0 → x = 5/3
x - 1 = 0 → x = 1

✔ Solutions: x = 1, x = 5/3

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Final Answer:
x = -8, -4; x = -2 ± √13; x = 1, 5/3