−18(2 sin2(t) + sin(t) − 1),which factors to f ′(t)

The given expression is −18(2 sin^2(t) + sin(t) − 1). To factor this, we first need to identify the common factors.

Step 1: The common factor here is -18. So, we can rewrite the expression as -18 * (2sin^2(t) + sin(t) - 1).

Step 2: Next, we look at the expression inside the parentheses. It's a quadratic expression in terms of sin(t). We can rewrite it as 2sin^2(t) + sin(t) - 1.

Step 3: Now, we need to factor this quadratic expression. We are looking for two numbers that multiply to -2 (the product of 2 and -1) and add up to 1 (the coefficient of sin(t)). The numbers that satisfy these conditions are 2 and -1.

Step 4: So, we can rewrite the quadratic expression as (2sin(t) - 1)(sin(t) + 1).

Step 5: Substitute this back into the original expression to get the factored form: -18 * (2sin(t) - 1)(sin(t) + 1).

So, the expression −18(2 sin^2(t) + sin(t) − 1) factors to -18 * (2sin(t) - 1)(sin(t) + 1).