<PRE><B>Question 243710</B>
{{{-45p^2 - 12p + 12p^3}}} Start with the given expression.



{{{12p^3-45p^2 - 12p }}} Rearrange the terms in descending degree.



{{{3p(4p^2-15p-4)}}} Factor out the GCF {{{3p}}}.



Now let&#39;s try to factor the inner expression {{{4p^2-15p-4}}}



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Looking at the expression {{{4p^2-15p-4}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{-15}}}, and the last term is {{{-4}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{-4}}} to get {{{(4)(-4)=-16}}}.



Now the question is: what two whole numbers multiply to {{{-16}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-15}}}?



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 {{{-16}}} (the previous product).



Factors of {{{-16}}}:

1,2,4,8,16

-1,-2,-4,-8,-16



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-16}}}.

1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-15}}}:



&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=red&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;1+(-16)=-15&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;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+(-8)=-6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+(-4)=0&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;16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+16=15&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;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+8=6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+4=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{1}}} and {{{-16}}} add to {{{-15}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-16}}} both multiply to {{{-16}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-15}}}



Now replace the middle term {{{-15p}}} with {{{p-16p}}}. Remember, {{{1}}} and {{{-16}}} add to {{{-15}}}. So this shows us that {{{p-16p=-15p}}}.



{{{4p^2+highlight(p-16p)-4}}} Replace the second term {{{-15p}}} with {{{p-16p}}}.



{{{(4p^2+p)+(-16p-4)}}} Group the terms into two pairs.



{{{p(4p+1)+(-16p-4)}}} Factor out the GCF {{{p}}} from the first group.



{{{p(4p+1)-4(4p+1)}}} 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.



{{{(p-4)(4p+1)}}} Combine like terms. Or factor out the common term {{{4p+1}}}



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So {{{3p(4p^2-15p-4)}}} then factors further to {{{3p(p-4)(4p+1)}}}



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Answer:



So {{{-45p^2 - 12p + 12p^3}}} completely factors to {{{3p(p-4)(4p+1)}}}.



In other words, {{{-45p^2 - 12p + 12p^3=3p(p-4)(4p+1)}}}.



Note: you can check the answer by expanding {{{3p(p-4)(4p+1)}}} to get {{{-45p^2 - 12p + 12p^3}}} or by graphing the original expression and the answer (the two graphs should be identical).
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