Question 756174
The description you give of the process is like an attempt to factorize the given quadratic expression.  By just looking at the polynomial {{{m^2+18m+77}}}, we might not know if it uses complex roots or complex binomial factors; maybe it does.  


So you want two numbers, like u and v so that uv=77 and u+v=18.  The idea is that {{{(m+u)(m+v)=m^2+18m+77}}}.  We will still get another quadratic equation in trying to find u and v.


Try first, all the factorizations for 77.  What are they?
{{{77=7*11}}} and {{{77=1*77}}}.  The second one does not promise much help.  Let's try seeing what the first one will do for us:


Will the product be 77 and the sum be 18?

{{{7*11=77}}} as we expect. 

{{{7+11=18}}} which is what we also wanted.

The numbers for u and v then are {{{u=7}}} and {{{v=11}}}.


The factorization for your given quadratic expression is {{{highlight((m+7)(m+11))}}}.


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:::   You WERE trying to put entries into a grid arrangment, so let's look at that.




_________m___________(___)________
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m_______m^2__________(___)____
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(__)____(___)_________77______



And maybe you are trying to fill in like this?....


_________m___________(u_)________
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m_______m^2__________(u*m_)____
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(v)____(v*m_)_________77______


That might help you to see how the terms occur in the multiplication process but you would still need to show combining the terms of m so as to say um+vm=18m which is how you knew that you want u+v=18.