<PRE><B>Question 215064</B>

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{{{h^4+2h^3-8h^2}}} Start with the given expression.



{{{h^2(h^2+2h-8)}}} Factor out the GCF {{{h^2}}}.



Now let&#39;s try to factor the inner expression {{{h^2+2h-8}}}



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Looking at the expression {{{h^2+2h-8}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{2}}}, and the last term is {{{-8}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-8}}} to get {{{(1)(-8)=-8}}}.



Now the question is: what two whole numbers multiply to {{{-8}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{2}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{-8}}} (the previous product).



Factors of {{{-8}}}:

1,2,4,8

-1,-2,-4,-8



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-8}}}.

1*(-8) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{2}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-8)=-7&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-4)=-2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+8=7&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+4=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{4}}} add to {{{2}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{4}}} both multiply to {{{-8}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{2}}}



Now replace the middle term {{{2h}}} with {{{-2h+4h}}}. Remember, {{{-2}}} and {{{4}}} add to {{{2}}}. So this shows us that {{{-2h+4h=2h}}}.



{{{h^2+highlight(-2h+4h)-8}}} Replace the second term {{{2h}}} with {{{-2h+4h}}}.



{{{(h^2-2h)+(4h-8)}}} Group the terms into two pairs.



{{{h(h-2)+(4h-8)}}} Factor out the GCF {{{h}}} from the first group.



{{{h(h-2)+4(h-2)}}} Factor out {{{4}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(h+4)(h-2)}}} Combine like terms. Or factor out the common term {{{h-2}}}



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So {{{h^2(h^2+2h-8)}}} then factors further to {{{h^2(h+4)(h-2)}}}



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Answer:



So {{{h^4+2h^3-8h^2}}} completely factors to {{{h^2(h+4)(h-2)}}}.



In other words, {{{h^4+2h^3-8h^2=h^2(h+4)(h-2)}}}.



Note: you can check the answer by expanding {{{h^2(h+4)(h-2)}}} to get {{{h^4+2h^3-8h^2}}} or by graphing the original expression and the answer (the two graphs should be identical).


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