Answer :
We start with the expression
[tex]$$
\frac{5x^6 - 15x^5 + 45x^2}{-5x^4}.
$$[/tex]
Step 1. Factor the numerator
Notice that each term in the numerator has a common factor of [tex]$5x^2$[/tex]. Factor it out:
[tex]$$
5x^6 - 15x^5 + 45x^2 = 5x^2 \left(x^4 - 3x^3 + 9\right).
$$[/tex]
Thus, the expression becomes
[tex]$$
\frac{5x^2 \left(x^4 - 3x^3 + 9\right)}{-5x^4}.
$$[/tex]
Step 2. Simplify by canceling common factors
First, cancel the factor of [tex]$5$[/tex]:
[tex]$$
\frac{5x^2 \left(x^4 - 3x^3 + 9\right)}{-5x^4} = \frac{x^2 \left(x^4 - 3x^3 + 9\right)}{-x^4}.
$$[/tex]
Next, simplify the power of [tex]$x$[/tex]. Notice that:
[tex]$$
\frac{x^2}{x^4} = \frac{1}{x^2}.
$$[/tex]
Thus, we have
[tex]$$
\frac{x^2 \left(x^4 - 3x^3 + 9\right)}{-x^4} = -\frac{x^4 - 3x^3 + 9}{x^2}.
$$[/tex]
Step 3. Write the expression in an expanded form
We can rewrite the simplified expression by splitting the fraction:
[tex]$$
-\frac{x^4}{x^2} + \frac{3x^3}{x^2} - \frac{9}{x^2}.
$$[/tex]
Simplify each term separately:
1. [tex]$\displaystyle \frac{x^4}{x^2} = x^2$[/tex],
2. [tex]$\displaystyle \frac{3x^3}{x^2} = 3x$[/tex],
3. [tex]$\displaystyle \frac{9}{x^2}$[/tex] remains the same.
Including the negative sign we obtain
[tex]$$
-x^2 + 3x - \frac{9}{x^2}.
$$[/tex]
Final Answer
The simplified form of the given expression is
[tex]$$
\boxed{-x^2 + 3x - \frac{9}{x^2}}.
$$[/tex]
[tex]$$
\frac{5x^6 - 15x^5 + 45x^2}{-5x^4}.
$$[/tex]
Step 1. Factor the numerator
Notice that each term in the numerator has a common factor of [tex]$5x^2$[/tex]. Factor it out:
[tex]$$
5x^6 - 15x^5 + 45x^2 = 5x^2 \left(x^4 - 3x^3 + 9\right).
$$[/tex]
Thus, the expression becomes
[tex]$$
\frac{5x^2 \left(x^4 - 3x^3 + 9\right)}{-5x^4}.
$$[/tex]
Step 2. Simplify by canceling common factors
First, cancel the factor of [tex]$5$[/tex]:
[tex]$$
\frac{5x^2 \left(x^4 - 3x^3 + 9\right)}{-5x^4} = \frac{x^2 \left(x^4 - 3x^3 + 9\right)}{-x^4}.
$$[/tex]
Next, simplify the power of [tex]$x$[/tex]. Notice that:
[tex]$$
\frac{x^2}{x^4} = \frac{1}{x^2}.
$$[/tex]
Thus, we have
[tex]$$
\frac{x^2 \left(x^4 - 3x^3 + 9\right)}{-x^4} = -\frac{x^4 - 3x^3 + 9}{x^2}.
$$[/tex]
Step 3. Write the expression in an expanded form
We can rewrite the simplified expression by splitting the fraction:
[tex]$$
-\frac{x^4}{x^2} + \frac{3x^3}{x^2} - \frac{9}{x^2}.
$$[/tex]
Simplify each term separately:
1. [tex]$\displaystyle \frac{x^4}{x^2} = x^2$[/tex],
2. [tex]$\displaystyle \frac{3x^3}{x^2} = 3x$[/tex],
3. [tex]$\displaystyle \frac{9}{x^2}$[/tex] remains the same.
Including the negative sign we obtain
[tex]$$
-x^2 + 3x - \frac{9}{x^2}.
$$[/tex]
Final Answer
The simplified form of the given expression is
[tex]$$
\boxed{-x^2 + 3x - \frac{9}{x^2}}.
$$[/tex]
