What are the factors of 2x + 3x - 54? Select two optionsN2x-92x-62x + 6DX-6x+6

Answer:The answer: The factors are (2x-9) and (x+6).The Problem:Factor [tex]2x^2+3x-54[/tex]Step-by-step explanation:So I'm going to do trial factors using the choices to aid me.Factored form for this problem if it exist will be in the form:[tex](mx+n)(kx+p)[/tex].In general this is what it would look like if we factored any quadratic in terms of [tex]x[/tex] (given the quadratic is not prime but technically you could factor even over the complex numbers).Let's look at:[tex](mx+n)(kx+p)[/tex]We want to choose [tex]k \text{ and } m[/tex] such that when you multiply them you get 2.  Well those would have to be 2 and 1.[tex](2x+n)(x+p)[/tex]we want to choose [tex]n \text{ and } p[/tex] such that when you multiply them you get -54. Based on the choices we want to get with -9 and 6, or 9 and -6. We don't know the order we want to choose it in either.For example which of these would work:[tex](2x-6)(x+9)[/tex][tex](2x+6)(x-9)[/tex][tex](2x-9)(x+6)[/tex][tex](2x+9)(x-6)[/tex]We are going to consider only the outer and inner of FOIL since we already know the first times the first is [tex]2x^2[/tex] and the last times the last is [tex]-54[/tex].Let's test the first one:[tex](2x-6)(x+9)[/tex]Outer:  2x(9)=18xInner: -6(x)=-6x------------------------ADD!18x-6x=12xThe first choice did not give us the middle term 3x.Trying the second one would give us the opposite since they are in the same form as previous just the + and - are switched.Let's look at the third one:[tex](2x-9)(x+6)[/tex]Outer: 2x(6)=12xInner: -9(x)=-9x--------------------------ADD!12x-9x=3xThis is the winner.The answer: The factors are (2x-9) and (x+6).