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Factor the trinomial completely.

9x^2 - 9x - 4

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To factor the trinomial \( 9x^2 - 9x - 4 \), we can use the method of factoring by grouping or the AC method.

  1. Identify the terms:

    • The coefficient \( a = 9 \), \( b = -9 \), and \( c = -4 \).

  2. Calculate \( ac \): \[ ac = 9 \times (-4) = -36 \]

  3. Look for two numbers that multiply to \( -36 \) and add up to \( -9 \). The appropriate pair is \( -12 \) and \( 3 \) because: \[ -12 + 3 = -9 \] \[ -12 \times 3 = -36 \]

  4. Rewrite the middle term using the two numbers: \[ 9x^2 - 12x + 3x - 4 \]

  5. Group the terms: \[ (9x^2 - 12x) + (3x - 4) \]

  6. Factor out the common factors: \[ 3x(3x - 4) + 1(3x - 4) \]

  7. Factor out the common binomial: \[ (3x - 4)(3x + 1) \]

So, the completely factored form of the trinomial \( 9x^2 - 9x - 4 \) is: \[ \boxed{(3x - 4)(3x + 1)} \]