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Before we dive into the actual process of factoring a trinomial, we need to understand the concept of the Greatest Common Factor (GCF). The GCF is the largest number that divides all terms in an algebraic expression evenly.
To find the GCF of a trinomial, follow these steps:

• Identify the coefficients of each term. For example, in the trinomial \[ 6a^3 + 12a^2 - 90a \], the coefficients are 6, 12, and -90.
• Find the GCF of these coefficients. Here, it is 6 because 6 is the largest number that divides 6, 12, and -90 without leaving a remainder.
• Examine the variable terms. In this case, all terms contain the variable 'a' in varying powers: \[ a^3, a^2, \text{and} \ a \]. The lowest power of 'a' present in each term is 'a'.

Thus, the GCF for \[6a^3 + 12a^2 - 90a\] is \[6a\].
Once the GCF is identified, it can be factored out from each term in the trinomial.

Trinomial factoring involves breaking down a trinomial into simpler expressions that multiply together to give the original trinomial. First, let's start by factoring out the Greatest Common Factor (GCF):
Given the trinomial \[6a^3 + 12a^2 - 90a\], and knowing the GCF is \[6a\], we factor it out:

• Divide each term by the GCF:

\[ \frac{6a^3}{6a} + \frac{12a^2}{6a} - \frac{90a}{6a} \]
This simplifies to:

• \[ a^2 + 2a - 15 \]

So the trinomial now looks like this:
\[ 6a(a^2 + 2a - 15) \]
The next step is to factor the quadratic expression within the parentheses. Look for two numbers that multiply to -15 and add up to 2. Those numbers are 5 and -3. Hence:

• \[ a^2 + 2a - 15 = (a + 5)(a - 3) \]

Now substitute back to get the final factored form:

• \[ 6a(a + 5)(a - 3) \]

This is the factored form of the original trinomial.

Understanding algebraic expressions is key to mastering trinomial factoring. An algebraic expression is a combination of variables, constants, and arithmetic operations. Here are some quick points to grasp:

• Variables represent unknown values and are usually denoted by letters like 'a', 'b', 'x', etc.
• Constants are fixed numbers like 1, 3, -5, etc.
• Expressions can include addition, subtraction, multiplication, and division.

A trinomial is a specific type of algebraic expression that consists of three terms.
In the trinomial \[ 6a^3 + 12a^2 - 90a \],

• 6, 12, and -90 are the coefficients.
• \