Final answer:
To factor the expression 3x^2 + 4x - 4 completely, find two numbers that multiply to -12 and add to 4, which are 6 and -2. Use them to split the middle term and factor by grouping, resulting in the final factored form (3x - 2)(x + 2).
Step-by-step explanation:
To factor completely the quadratic expression 3x2 + 4x - 4, we are looking for two binomials that when multiplied together give us the original expression. We need to find two numbers that multiply to give us the product of the coefficient of the x2 term (which is 3) and the constant term (which is -4), that is, -12, and those two numbers must also add up to give us the coefficient of the x term (which is 4).
We find that these two numbers are 6 and -2. Now we write the middle term, 4x, as a sum of two terms using 6 and -2: 3x2 + 6x - 2x - 4. Next, we group the terms: (3x2 + 6x) - (2x + 4). Factoring by grouping, we get 3x(x + 2) - 2(x + 2), and we can factor out the common binomial factor (x + 2), resulting in (3x - 2)(x + 2) as the completely factored form of the quadratic expression.