Question

Factor completely.
20m^3-10m-5m^2+10m^4

Answer 1

To factor the expression \(20m^3 - 10m - 5m^2 + 10m^4\), we can start by rearranging the terms in descending order of the powers of \(m\):

\[
10m^4 + 20m^3 - 5m^2 - 10m
\]

Next, we will group the terms for easier factoring:

\[
(10m^4 + 20m^3) + (-5m^2 - 10m)
\]

Now, we can factor out the common factors from each group:

1. From the first group \(10m^4 + 20m^3\), we can factor out \(10m^3\):
\[
10m^3(m + 2)
\]

2. From the second group \(-5m^2 - 10m\), we can factor out \(-5m\):
\[
-5m(m + 2)
\]

So, we rewrite the entire expression:

\[
10m^3(m + 2) - 5m(m + 2)
\]

Now, notice that we have a common factor of \((m + 2)\). We can factor that out:

\[
(m + 2)(10m^3 - 5m)
\]

Now, we can factor the second term \(10m^3 - 5m\) further. Notice that we can factor out \(5m\):

\[
5m(2m^2 - 1)
\]

Finally, substituting back into our expression, we have:

\[
(m + 2)(5m(2m^2 - 1))
\]

Thus, the completely factored form of the expression \(20m^3 - 10m - 5m^2 + 10m^4\) is:

\[
5m(m + 2)(2m^2 - 1)
\]