Answer :
We start with the expression
[tex]$$12x^4 + 18x^3 + 45x^2.$$[/tex]
Step 1. Find the common numerical factor.
The coefficients are 12, 18, and 45. Their greatest common divisor (GCD) is 3. Therefore, we factor out 3.
Step 2. Factor out the common variable factor.
Each term of the expression contains a power of [tex]$x$[/tex], and the smallest power is [tex]$x^2$[/tex]. This means we can also factor out [tex]$x^2$[/tex].
Step 3. Combine the common factors.
Factoring [tex]$3x^2$[/tex] out of the expression gives:
[tex]$$12x^4 + 18x^3 + 45x^2 = 3x^2\left(\frac{12x^4}{3x^2} + \frac{18x^3}{3x^2} + \frac{45x^2}{3x^2}\right).$$[/tex]
Simplify inside the parentheses:
[tex]\[
\begin{aligned}
\frac{12x^4}{3x^2} &= 4x^2, \\
\frac{18x^3}{3x^2} &= 6x, \\
\frac{45x^2}{3x^2} &= 15.
\end{aligned}
\][/tex]
So the expression becomes
[tex]$$3x^2(4x^2 + 6x + 15).$$[/tex]
Final Answer:
The completely factored form of the polynomial is
[tex]$$\boxed{3x^2(4x^2 + 6x + 15)}.$$[/tex]
[tex]$$12x^4 + 18x^3 + 45x^2.$$[/tex]
Step 1. Find the common numerical factor.
The coefficients are 12, 18, and 45. Their greatest common divisor (GCD) is 3. Therefore, we factor out 3.
Step 2. Factor out the common variable factor.
Each term of the expression contains a power of [tex]$x$[/tex], and the smallest power is [tex]$x^2$[/tex]. This means we can also factor out [tex]$x^2$[/tex].
Step 3. Combine the common factors.
Factoring [tex]$3x^2$[/tex] out of the expression gives:
[tex]$$12x^4 + 18x^3 + 45x^2 = 3x^2\left(\frac{12x^4}{3x^2} + \frac{18x^3}{3x^2} + \frac{45x^2}{3x^2}\right).$$[/tex]
Simplify inside the parentheses:
[tex]\[
\begin{aligned}
\frac{12x^4}{3x^2} &= 4x^2, \\
\frac{18x^3}{3x^2} &= 6x, \\
\frac{45x^2}{3x^2} &= 15.
\end{aligned}
\][/tex]
So the expression becomes
[tex]$$3x^2(4x^2 + 6x + 15).$$[/tex]
Final Answer:
The completely factored form of the polynomial is
[tex]$$\boxed{3x^2(4x^2 + 6x + 15)}.$$[/tex]
