document.write( "Question 190281: Factor\r
\n" ); document.write( "\n" ); document.write( "4x^2+16x-48 \n" ); document.write( "

Algebra.Com's Answer #142807 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2B16x-48\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"4%28x%5E2%2B4x-12%29\"$ Factor out the GCF $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2%2B4x-12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2%2B4x-12\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"-12\"$ where the coefficients are 1 and -12 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -12 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
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\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 $\"x%5E2%2B4x-12\"$ , replace $\"4x\"$ with $\"-2x%2B6x\"$ (notice $\"-2x%2B6x\"$ adds up to $\"4x\"$ . So it is equivalent to $\"4x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-2x%2B6x%29-12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2-2x%2B6x-12\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-2x%29%2B%286x-12%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2%29%2B6%28x-2%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"6\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B6%29%28x-2%29\"$ Since we have a common term of $\"x-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-2x%2B6x-12\"$ factors to $\"%28x%2B6%29%28x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2%2B4x-12\"$ factors to $\"%28x%2B6%29%28x-2%29\"$ (since $\"x%5E2%2B4x-12\"$ is equivalent to $\"x%5E2-2x%2B6x-12\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4%28x%5E2%2B4x-12%29\"$ and factors further to $\"4%28x%2B6%29%28x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2%2B16x-48\"$ completely factors to $\"4%28x%2B6%29%28x-2%29\"$ \r
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