Answer :

We start with the polynomial

[tex]$$30x^3 - 45x^2 + 15x.$$[/tex]

Step 1. Factor out the Greatest Common Factor (GCF):

All the terms share a common factor of [tex]$15x$[/tex]. Factoring this out, we have

[tex]$$30x^3 - 45x^2 + 15x = 15x(2x^2 - 3x + 1).$$[/tex]

Step 2. Factor the Quadratic [tex]$2x^2 - 3x + 1$[/tex]:

We now focus on factoring the quadratic expression:

[tex]$$2x^2 - 3x + 1.$$[/tex]

To factor a quadratic of the form [tex]$ax^2+bx+c$[/tex], we look for two numbers that multiply to [tex]$(a \cdot c)$[/tex] and add to [tex]$b$[/tex]. In this case:
- [tex]$a \cdot c = 2 \cdot 1 = 2$[/tex], and
- [tex]$b = -3$[/tex].

The two numbers that satisfy these conditions are [tex]$-1$[/tex] and [tex]$-2$[/tex], since [tex]$(-1) \cdot (-2) = 2$[/tex] and [tex]$(-1) + (-2) = -3$[/tex].

Using these numbers, we can rewrite the middle term and factor by grouping:

[tex]\[
\begin{align*}
2x^2 - 3x + 1 &= 2x^2 - 2x - x + 1 \\
&= (2x^2 - 2x) - (x - 1) \\
&= 2x(x - 1) - 1(x - 1) \\
&= (x - 1)(2x - 1).
\end{align*}
\][/tex]

Step 3. Write the Fully Factored Form:

Substitute the factored quadratic back into the expression:

[tex]$$15x(2x^2 - 3x + 1) = 15x\,(x - 1)(2x - 1).$$[/tex]

Thus, the fully factored expression is:

[tex]$$\boxed{15x(x - 1)(2x - 1)}.$$[/tex]

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