Answer:
[tex]4m^4+36m^3+96m^2+80m= 4m\left(m+2\right)^2\left(m+5\right)[/tex].
Step-by-step explanation:
To factor the polynomial [tex]4m^4\:+\:36m^3\:+\:96m^2\:+\:80m[/tex] you must:
Factor out common term [tex]4m[/tex]:
[tex]4m\left(m^3+9m^2+24m+20\right)[/tex]
Next, factor [tex]m^3+9m^2+24m+20[/tex].
Since all coefficients are integers, apply the Rational Zeros Theorem.
The Rational Zero Theorem states that, if the polynomial [tex]f\left(x\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[/tex] has integer coefficients, then every rational zero of [tex]f(x)[/tex] has the form [tex]\frac{p}{q}[/tex] where [tex]p[/tex] is a factor of the constant term [tex]a_0[/tex] and [tex]q[/tex] is a factor of the leading coefficient [tex]a_n[/tex]
Using the Rational Zero Theorem,
[tex]a_0=20,\:\quad a_n=1[/tex]
The dividers of [tex]a_0[/tex]: [tex]1,\:2,\:4,\:5,\:10,\:20[/tex]
The dividers of [tex]a_n[/tex] : 1
Therefore, check the following rational numbers: [tex]\pm \frac{1,\:2,\:4,\:5,\:10,\:20}{1}[/tex]
[tex]-\frac{2}{1}[/tex] is a root of the expression, so factor out [tex]m+2[/tex]
[tex]m^3+9m^2+24m+20=\left(m+2\right)\frac{m^3+9m^2+24m+20}{m+2}[/tex]
[tex]\frac{m^3+9m^2+24m+20}{m+2}=m^2+7m+10[/tex]
Next, factor [tex]m^2+7m+10[/tex]
[tex]m^2+7m+10= \left(m+2\right)\left(m+5\right)[/tex]
Therefore,
[tex]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)[/tex].
