```Question 307009
Let x= the integer

The equation is
{{{2x+3x^2 = 261}}}

This is a quadratic equation, so you have to set it equal to zero:
{{{3x^2 + 2x - 261=0}}}

This is a difficult factoring problem, since 261 is a rather large number.  If you happen to notice that it is divisible by 9, it turns out that 261=9*29, which turns out to be the correct factoring combination.

{{{(3x-29)(x-9)= 0}}}

The first solution is NOT an integer, so you need not solve this.  The second and most obvious solution is x=9, the ONLY integer solution.

Check:  Take "twice 9 + 3 times it's square", and you get
2*9 + 3*9^2
18 + 3*81
18+243
261  --> It checks!!

I do have some resources on my website that might be helpful in solving this question.  In particular, the hardest part is the factoring of this horrible trinomial!!  You can find my website by doing a Google search for my name "Dr. Robert J. Rapalje".  Look for "Rapalje Homepage", which should be the first item on the search.  Once you find my homepage, for a written explanation, click on "Basic, Intermediate, and College Algebra: One Step at a Time."  Look in Basic Algebra in Chapter 2 for the section on Advanced Trinomial Factoring.  There you will find SHORTCUTS to help you factor these problems, and detailed explanations, and solutions to problems in LIVING COLOR!!

To see my FREE video explanation, look for the link on my homepage called "Rapalje Videos in Living Color."  This time, choose Intermediate Algebra, and look for the topic of "Factoring", in particular the part of the video on "Advanced Trinomial Factoring."

I have many other topics explained in this way from Basic, Intermediate, and College Algebra.  Please check out my website to see what I have.  It's all FREE, and if you send me an Email, I'll be glad to help you find something.  I'm not selling anything.  If you get a good explanation, math is usually not as hard as you think it is.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus```