Question

completely.

1. $3x^3 - 3x^2 - 60x$

Explanation:

Step1: Factor out the GCF

First, we identify the greatest common factor (GCF) of the terms \(3x^3\), \(-3x^2\), and \(-60x\). The GCF of the coefficients \(3\), \(-3\), and \(-60\) is \(3\), and the GCF of the variable terms \(x^3\), \(x^2\), and \(x\) is \(x\). So, we factor out \(3x\) from each term:
$$3x^3 - 3x^2 - 60x = 3x(x^2 - x - 20)$$

Step2: Factor the quadratic trinomial

Next, we factor the quadratic trinomial \(x^2 - x - 20\). We need to find two numbers that multiply to \(-20\) (the constant term) and add up to \(-1\) (the coefficient of the middle term). The numbers \(-5\) and \(4\) satisfy these conditions because \((-5) \times 4 = -20\) and \(-5 + 4 = -1\). So, we can factor \(x^2 - x - 20\) as \((x - 5)(x + 4)\):
$$x^2 - x - 20 = (x - 5)(x + 4)$$

Step3: Combine the factors

Putting it all together, we substitute the factored form of the quadratic trinomial back into the expression from Step 1:
$$3x(x^2 - x - 20) = 3x(x - 5)(x + 4)$$

Answer:

\(3x(x - 5)(x + 4)\)