Question 258816

<pre><font size = 4 color = "indigo"><b>
{{{-4x^2 + 23x - 28}}}

First factor out a -1

{{{-(red(4)x^2-23x+red(28))}}}

Multiply the {{{red(4)}}} by the {{{red(28)}}}

Get 112.

Write down all the ways to break 112 into 2 factors of whole numbers

112x1
 56x2
 28x4
 16x7
 14x8

Notice that the last sign is a plus:

{{{"-("}}}{{{4x^2-23x}}}{{{red("+")}}}{{{28)}}}

Plus means add so beside those numbers we
just multiplied, let's add them:

112x1  112+1=113
 56x2   56+2=58 
 28x4   28+4=32
 16x7   16+7=23
 14x8   14+8=22

Look at the middle term of

{{{-(4x^2-red(23)x+28)}}}

in absolute value.  It is a 23.  Now 
look among those sums we just found.  
Is there a 23 among them?  Yes.  That
means it can be factored. We find:

 16x7   16+7=23

So let's replace 23 by {{{red((16+7))}}}

{{{-(4x^2-red((16+7))x+28)}}}

Now distribute it out

{{{-(4x^2-16x-7x+28)}}}

Factor 4x out of the first two terms:

{{{-(4x(x-4)-7x+28)}}}

Factor -7 out of the last two terms:

{{{-(4x(x-4)-7(x-4))}}}

Notice the common factor {{{green((x-4)))}}}


{{{-(4x*green((x-4))-7*green((x-4)))}}}

Factor out the green {{{green((x-4))}}}

{{{-(green((x-4))(4x-7))}}}

We can now remove the outer parentheses but leave
the negative sign in front:

{{{-(x-4)(4x-7)}}}

Edwin</pre>