Let's start with the given expression:
[tex]\[
\frac{-70 w^2 + 70}{w^2 - 5w + 4}
\][/tex]
First, consider the numerator and factor it where possible. The numerator is:
[tex]\[
-70 w^2 + 70
\][/tex]
Factor out the common factor of [tex]\( -70 \)[/tex]:
[tex]\[
-70 (w^2 - 1)
\][/tex]
Notice that [tex]\( w^2 - 1 \)[/tex] is a difference of squares and can be further factored:
[tex]\[
w^2 - 1 = (w + 1)(w - 1)
\][/tex]
Thus, the numerator becomes:
[tex]\[
-70 (w + 1)(w - 1)
\][/tex]
Next, consider the denominator:
[tex]\[
w^2 - 5w + 4
\][/tex]
We need to factor this quadratic expression. To do this, look for factors of 4 that add up to -5. The factors -1 and -4 work:
[tex]\[
w^2 - 5w + 4 = (w - 1)(w - 4)
\][/tex]
Now we have:
[tex]\[
\frac{-70 (w + 1)(w - 1)}{(w - 1)(w - 4)}
\][/tex]
We can cancel the common factor of [tex]\( (w - 1) \)[/tex] in the numerator and the denominator, assuming [tex]\( w \neq 1 \)[/tex]:
[tex]\[
\frac{-70 (w + 1)}{(w - 4)}
\][/tex]
Therefore, the completely factored and simplified form of the given expression is:
[tex]\[
\boxed{\frac{-70 (w + 1)}{(w - 4)}}
\][/tex]
