To simplify the rational expression
\[ \frac{-5t^2 + 10t^3 - 5t^4}{6t^3 - 12t^4 + 6t^5}, \]
we will first factor both the numerator and the denominator.
Step 1: Factor the Numerator
The numerator is: \[ -5t^2 + 10t^3 - 5t^4. \] We can factor out \(-5t^2\): \[ -5t^2(1 - 2t + t^2). \] Next, we will factor the quadratic \(1 - 2t + t^2\). This can be rewritten as: \[ t^2 - 2t + 1 = (t - 1)^2. \] Thus, the numerator can be factored as: \[ -5t^2(t - 1)^2. \]
Step 2: Factor the Denominator
Now for the denominator: \[ 6t^3 - 12t^4 + 6t^5. \] We can factor out \(6t^3\): \[ 6t^3(1 - 2t + t^2). \] As we noted before, \(1 - 2t + t^2 = (t - 1)^2\). So we can rewrite the denominator as: \[ 6t^3(t - 1)^2. \]
Step 3: Write the Factored Form of the Rational Expression
Now substituting the factored forms back into the rational expression, we get: \[ \frac{-5t^2(t - 1)^2}{6t^3(t - 1)^2}. \]
Step 4: Cancel Common Factors
We can cancel the \((t - 1)^2\) terms from the numerator and denominator: \[ \frac{-5t^2}{6t^3}. \] We also notice that \(t^2\) in the numerator and \(t^3\) in the denominator share a common factor of \(t^2\): \[ \frac{-5}{6t}. \]
Final Result
Thus, the rational expression simplified to lowest terms is: \[ \frac{-5}{6t}. \]