The quadratic equation 4sin²θ - 4sinθ - 3 can be factored into (2sinθ + 1)(2sinθ - 3), which can be used to find the value(s) of θ when set equal to zero.
The student is looking to factor the quadratic equation 4sin²θ - 4sinθ - 3. To factor this, we treat it like a standard quadratic by setting up two binomials (assume sinθ is just a variable like 'x'). We look for two numbers that multiply to (4)(-3) = -12 and add up to -4, the coefficient of the middle term. After finding those numbers, we can factor the quadratic.
Let's try the factors of -12 and see which pair adds up to -4. The pairs of factors are (-1, 12), (1, -12), (-2, 6), and (2, -6). The pair that adds up to -4 is (2, -6). Thus, we write: (4sinθ - 6)(sinθ + 1) = 0
So, the factored form of 4sin²θ - 4sinθ - 3 is (2sinθ + 1)(2sinθ - 3), which can be solved for θ by setting each factor equal to zero.