Question

1. $15a^3 + 10a^2 - 40a$

Explanation:

Step1: Find the GCF of coefficients and variable

The coefficients are 15, 10, -40. The GCF of 15, 10, 40 is 5. For the variable part, the lowest power of \(a\) is \(a^1\) (since \(a^3\), \(a^2\), \(a\) have \(a\) with power 1 as the minimum). So the GCF is \(5a\).

Step2: Factor out the GCF

Divide each term by \(5a\):
\(\frac{15a^3}{5a} = 3a^2\), \(\frac{10a^2}{5a} = 2a\), \(\frac{-40a}{5a} = -8\).
So, \(15a^3 + 10a^2 - 40a = 5a(3a^2 + 2a - 8)\).
(We can further factor \(3a^2 + 2a - 8\) by finding two numbers that multiply to \(3\times(-8)= -24\) and add to 2. Those numbers are 6 and -4. Then \(3a^2 + 6a - 4a - 8 = 3a(a + 2) - 4(a + 2) = (3a - 4)(a + 2)\). So the fully factored form is \(5a(3a - 4)(a + 2)\), but the first step of factoring out the GCF is the main part here.)

Answer:

\(5a(3a^2 + 2a - 8)\) (or \(5a(3a - 4)(a + 2)\) for full factorization)