[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]
[tex] \large \boxed{\mathbb{ANSWER\: WITH\: EXPLANATION} \downarrow}[/tex]
By.. 4m exponent 2, I'm going to assume that you meant this ⇨ 4m². If so, then here's how you should solve your question. I've included 2 ways of solving the question. You can choose your method. I'd suggest the 2nd one since it's a whole lot easier than the first method.
Method 1 :-
[tex] \tt \: 4m ^ { 2 } +5m \\ [/tex]
Quadratic polynomial can be factored using the transformation [tex]\tt\: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)[/tex], where [tex]\tt\:x_{1} and x_{2}[/tex] are the solutions of the quadratic equation ax²+bx+c=0.
[tex] \tt \: 4m^{2}+5m=0 [/tex]
All equations of the form ax²+bx+c=0 can be solved using the quadratic formula: [tex]\tt\frac{-b±\sqrt{b^{2}-4ac}}{2a}\\ [/tex]. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
[tex] \tt \: m=\frac{-5±\sqrt{5^{2}}}{2\times 4} \\ [/tex]
Take the square root of 5².
[tex] \tt \: m=\frac{-5±5}{2\times 4} \\ [/tex]
Multiply 2 times 4.
[tex] \tt \: m=\frac{-5±5}{8} \\ [/tex]
Now solve the equation m=[tex]\tt\frac{-5±5}{8}[/tex] when ± is plus. Add -5 to 5.
[tex] \tt \: m=\frac{0}{8} \\ \\ \tt \: m = 0[/tex]
Now solve the equation m=[tex]\tt\frac{-5±5}{8} [/tex] when ± is minus. Subtract 5 from -5.
[tex] \tt \: m=\frac{-10}{8} \\ [/tex]
Reduce the fraction -10/8 to its lowest terms by extracting and cancelling out 2.
Factor the original expression using [tex]\tt\:ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)[/tex]. Substitute 0 for [tex]\tt\:x_{1} \:and \:-\frac{5}{4} \:for\: x_{2}[/tex].
[tex] \tt \: 4m^{2}+5m=4m\left(m-\left(-\frac{5}{4}\right)\right) \\ [/tex]
Simplify all of the expressions of the form [tex]\tt\:p-\left(-q\right)[/tex] to p+q.
[tex] \tt \: 4m^{2}+5m=4m\left(m+\frac{5}{4}\right) \\ [/tex]
Add 5/4 to m by finding a common denominator and adding the numerators. Then reduce the fraction to its lowest terms if possible.
[tex] \tt \: 4m^{2}+5m=4m\times \left(\frac{4m+5}{4}\right) \\ [/tex]
Cancel out 4, the greatest common factor in 4 and 4.
[tex] \tt \: 4m^{2}+5m= \boxed{\boxed{\bf \: m\left(4m+5\right) }}[/tex]
Don't worry. If this process seems long & wears you out or if you haven't learned the biquadratic formula yet, you can just use the simple method of factoring out the common term (m). Here's how you do that [tex]\downarrow[/tex]
Method 2 :-
[tex] \tt \: 4m ^ { 2 } +5m[/tex]
Factor out m.
[tex] = \boxed{ \boxed{ \bf \: m\left(4m+5\right) }}[/tex]
See, the second method is easier. But, if your question comes for a lot of marks then you might prefer using the first method.
