Question

Factor the polynomial. 4m4 + 36m3 + 96m2 + 80m

Answer 1

Add them together then get answer

Answer 2

`4m^4+36m^3+96m^2+80m= 4m\left(m+2\right)^2\left(m+5\right)`.

Explanation:

To factor the polynomial
`4m^4\:+\:36m^3\:+\:96m^2\:+\:80m` you must:

Factor out common term
`4m`:

`4m\left(m^3+9m^2+24m+20\right)`

Next, factor
`m^3+9m^2+24m+20`.

Since all coefficients are integers, apply the Rational Zeros Theorem.

The Rational Zero Theorem states that, if the polynomial
`f\left(x\right)={a}_(n){x}^(n)+{a}_(n - 1){x}^(n - 1)+...+{a}_(1)x+{a}_(0)` has integer coefficients, then every rational zero of
`f(x)` has the form
`(p)/(q)` where
`p` is a factor of the constant term
`a_0` and
`q` is a factor of the leading coefficient
`a_n`

Using the Rational Zero Theorem,

`a_0=20,\:\quad a_n=1`

The dividers of
`a_0`:
`1,\:2,\:4,\:5,\:10,\:20`

The dividers of
`a_n` : 1

Therefore, check the following rational numbers:
`\pm (1,\:2,\:4,\:5,\:10,\:20)/(1)`

`-(2)/(1)` is a root of the expression, so factor out
`m+2`

`m^3+9m^2+24m+20=\left(m+2\right)(m^3+9m^2+24m+20)/(m+2)`

`(m^3+9m^2+24m+20)/(m+2)=m^2+7m+10`

Next, factor
`m^2+7m+10`

`m^2+7m+10= \left(m+2\right)\left(m+5\right)`

Therefore,

`4m^4+36m^3+96m^2+80m=4m\left(m+2\right)\left(m+2\right)\left(m+5\right)=4m\left(m+2\right)^2\left(m+5\right)`.