document.write( "Question 119790: factor 32a^2+4a-6 \n" ); document.write( "

Algebra.Com's Answer #87777 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"32a%5E2%2B4a-6\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2816a%5E2%2B2a-3%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"16a%5E2%2B2a-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"16a%5E2%2B2a-3\"$ we can see that the first term is $\"16a%5E2\"$ and the last term is $\"-3\"$ where the coefficients are 16 and -3 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 16 and the last coefficient -3 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient 2? Let's list all of the factors of -48:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -48:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...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 -48\r
\n" ); document.write( "\n" ); document.write( "(1)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(48)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(16)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(8)\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 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2\r
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First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and 8 add up to 2 and multiply to -48\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"16a%5E2%2B2a-3\"$ , replace $\"2a\"$ with $\"-6a%2B8a\"$ (notice $\"-6a%2B8a\"$ adds up to $\"2a\"$ . So it is equivalent to $\"2a\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"16a%5E2%2Bhighlight%28-6a%2B8a%29%2B-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"16a%5E2-6a%2B8a-3\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2816a%5E2-6a%29%2B%288a-3%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%288a-3%29%2B1%288a-3%29\"$ Factor out the GCF of $\"2a\"$ out of the first group. Factor out the GCF of $\"1\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282a%2B1%29%288a-3%29\"$ Since we have a common term of $\"8a-3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"16a%5E2-6a%2B8a-3\"$ factors to $\"%282a%2B1%29%288a-3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"16a%5E2%2B2a-3\"$ factors to $\"%282a%2B1%29%288a-3%29\"$ (since $\"16a%5E2%2B2a-3\"$ is equivalent to $\"16a%5E2-6a%2B8a-3\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"2%282a%2B1%29%288a-3%29\"$ Now reintroduce the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"32a%5E2%2B4a-6\"$ factors to $\"2%282a%2B1%29%288a-3%29\"$ \r
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