document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114249>Question 114249</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rewrite the middle term as the sum of two terms and then factor completely.\r<BR>\n" );
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document.write( "12w squared + 19w + 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83165 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=83165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12w%5E2+%2B+19w+%2B+4=%28aw%2Bb%29%28cw%2Bd%29 ALIGN=MIDDLE  ALT=\"12w%5E2+%2B+19w+%2B+4=%28aw%2Bb%29%28cw%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Find a,b,c,d using the FOIL method (First, Outer, Inner, Last) and equating to the coefficients of your equation.\r<BR>\n" );
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document.write( "Now to expand the equation you use the FOIL method (First, Outer, Inner, Last)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28aw%2Bb%29%28cw%2Bd%29 ALIGN=MIDDLE  ALT=\"%28aw%2Bb%29%28cw%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28highlight%28aw%29%2Bb%29%28highlight%28cw%29%2B%28d%29%29=%28aw%29%28cw%29=acw%5E2 ALIGN=MIDDLE  ALT=\"%28highlight%28aw%29%2Bb%29%28highlight%28cw%29%2B%28d%29%29=%28aw%29%28cw%29=acw%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Outer : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28highlight%28aw%29%2Bb%29%28cw%2Bhighlight%28d%29%29=%28aw%29%28d%29=adw ALIGN=MIDDLE  ALT=\"%28highlight%28aw%29%2Bb%29%28cw%2Bhighlight%28d%29%29=%28aw%29%28d%29=adw\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Inner : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28aw%2Bhighlight%28b%29%29%28highlight%28cw%29%2B%28d%29%29=%28b%29%28cw%29=bcw ALIGN=MIDDLE  ALT=\"%28aw%2Bhighlight%28b%29%29%28highlight%28cw%29%2B%28d%29%29=%28b%29%28cw%29=bcw\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Last : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28aw%2Bhighlight%28b%29%29%28cw%2Bhighlight%28d%29%29=%28b%29%28d%29=bd ALIGN=MIDDLE  ALT=\"%28aw%2Bhighlight%28b%29%29%28cw%2Bhighlight%28d%29%29=%28b%29%28d%29=bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28aw%2Bb%29%28cw%2Bd%29=%28ac%29%2Aw%5E2%2B%28ad%29%2Aw%2B%28bc%29w%2Bbd ALIGN=MIDDLE  ALT=\"%28aw%2Bb%29%28cw%2Bd%29=%28ac%29%2Aw%5E2%2B%28ad%29%2Aw%2B%28bc%29w%2Bbd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28aw%2Bb%29%28cw%2Bd%29=%28ac%29%2Aw%5E2%2B%28ad%2Bbc%29%2Aw%2Bbd ALIGN=MIDDLE  ALT=\"%28aw%2Bb%29%28cw%2Bd%29=%28ac%29%2Aw%5E2%2B%28ad%2Bbc%29%2Aw%2Bbd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Comparing to your equation <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12w%5E2+%2B+19w+%2B+4=%28ac%29%2Aw%5E2%2B%28ad%2Bbc%29%2Aw%2Bbd ALIGN=MIDDLE  ALT=\"12w%5E2+%2B+19w+%2B+4=%28ac%29%2Aw%5E2%2B%28ad%2Bbc%29%2Aw%2Bbd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ac=12 ALIGN=MIDDLE  ALT=\"ac=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ad%2Bbc=19 ALIGN=MIDDLE  ALT=\"ad%2Bbc=19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bd=4 ALIGN=MIDDLE  ALT=\"bd=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From 1, since ac=12, you can choose 12 and 1, 6 and 2, 4 and 3, and also 3 and 4, 2 and 6, and 1 and 12 for possible a-c combinations.<BR>\n" );
document.write( "From 3, since bd=4, you can choose 4 and 1, 2 and 2, and 1 and 4 for possible b-d combinations.<BR>\n" );
document.write( "Let's start with a=6 and c=2<BR>\n" );
document.write( "From 2, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6d%2B2b=19 ALIGN=MIDDLE  ALT=\"6d%2B2b=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Notice the coefficients of b and d. <BR>\n" );
document.write( "This solution set cannot be a solution because 2 even numbers will never sum to an odd number. <BR>\n" );
document.write( "So 6 and 2 and 2 and 6 are out as possible a-c combinations. <BR>\n" );
document.write( "Start again,<BR>\n" );
document.write( "Let's choose a=12 and c=1, <BR>\n" );
document.write( "From 2, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2Ad%2Bb=19 ALIGN=MIDDLE  ALT=\"12%2Ad%2Bb=19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now substitute possible b,d values.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A1%2B4=16 ALIGN=MIDDLE  ALT=\"12%2A1%2B4=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > for b=4, d=1.  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A2%2B2=26 ALIGN=MIDDLE  ALT=\"12%2A2%2B2=26\"   DISABLED_style= \"max-width:90%; height:auto;\" > for b=2, d=2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A4%2B1=49 ALIGN=MIDDLE  ALT=\"12%2A4%2B1=49\"   DISABLED_style= \"max-width:90%; height:auto;\" > for b=1, d=4.<BR>\n" );
document.write( "No solutions for a=12 and c=1. <BR>\n" );
document.write( "Our last chance is 4 and 3.<BR>\n" );
document.write( "Let's start with a=4 and c=3.<BR>\n" );
document.write( "From 2, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ad%2B3%2Ab=19 ALIGN=MIDDLE  ALT=\"4%2Ad%2B3%2Ab=19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again, substitute possible b,d values.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A1%2B3%2A4=16 ALIGN=MIDDLE  ALT=\"4%2A1%2B3%2A4=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > for b=4, d=1  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A2%2B3%2A2=14 ALIGN=MIDDLE  ALT=\"4%2A2%2B3%2A2=14\"   DISABLED_style= \"max-width:90%; height:auto;\" > for b=2, d=2 <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A4%2B3%2A1=19 ALIGN=MIDDLE  ALT=\"4%2A4%2B3%2A1=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > for b=1, d=4 <BR>\n" );
document.write( "Looks like a winner. <BR>\n" );
document.write( "a=4,c=3,b=1,d=4<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12w%5E2+%2B+19w+%2B+4=%284w%2B1%29%283w%2B4%29 ALIGN=MIDDLE  ALT=\"12w%5E2+%2B+19w+%2B+4=%284w%2B1%29%283w%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );