document.write( "Question 191754: Factor by grouping\r
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Algebra.Com's Answer #143873 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"21a%5E2-28ab-28b%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"7%283a%5E2-4ab-4b%5E2%29\"$ Factor out the GCF $\"7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"3a%5E2-4ab-4b%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"3a%5E2-4ab-4b%5E2\"$ we can see that the first term is $\"3a%5E2\"$ and the last term is $\"-4b%5E2\"$ where the coefficients are 3 and -4 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient -4 to get -12. Now what two numbers multiply to -12 and add to the middle coefficient -4? Let's list all of the factors of -12:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -12:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -12\r
\n" ); document.write( "\n" ); document.write( "(1)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-4)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(6)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(4)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4\r
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First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 2 and -6 add up to -4 and multiply to -12\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3a%5E2-4ab-4b%5E2\"$ , replace $\"-4ab\"$ with $\"2ab-6ab\"$ (notice $\"2ab-6ab\"$ adds up to $\"-4ab\"$ . So it is equivalent to $\"-4ab\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3a%5E2%2Bhighlight%282ab-6ab%29-4b%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3a%5E2%2B2ab-6ab-4b%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283a%5E2%2B2ab%29%2B%28-6ab-4b%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"a%283a%2B2b%29-2b%283a%2B2b%29\"$ Factor out the GCF of $\"a\"$ out of the first group. Factor out the GCF of $\"-2b\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a-2b%29%283a%2B2b%29\"$ Since we have a common term of $\"3a%2B2b\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3a%5E2%2B2ab-6ab-4b%5E2\"$ factors to $\"%28a-2b%29%283a%2B2b%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3a%5E2-4ab-4b%5E2\"$ factors to $\"%28a-2b%29%283a%2B2b%29\"$ (since $\"3a%5E2-4ab-4b%5E2\"$ is equivalent to $\"3a%5E2%2B2ab-6ab-4b%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"7%283a%5E2-4ab-4b%5E2%29\"$ and factors further to $\"7%28a-2b%29%283a%2B2b%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"21a%5E2-28ab-28b%5E2\"$ completely factors to $\"7%28a-2b%29%283a%2B2b%29\"$ \r
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