Question 1123541
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I would go about this in a different order....<br>
Given the root of -2, first use synthetic division to remove that root.<br><pre>

  -2 |  1  0  -11  -26  48  144
     |    -2    4   14  24 -144
     --------------------------
        1 -2   -7  -12  72   0
<br></pre>
At this point we know<br>
{{{m^5 - 11m^3-26m^2+48m+144 = (m+2)(m^4-2m^3-7m^2-12m+72)}}}<br>
Next, given the root -2+2i, we know -2-2i is also a root, because complex roots occur in conjugate pairs.<br>
We can get the quadratic factor corresponding to that pair of roots by using the fact that in the quadratic equation x^2+bx+c=0 the sum of the roots is -b and the product is c.<br>
The sum of these two roots is -4; their product is 4-4i^2 = 4+4 = 8.  So the quadratic factor corresponding to these two roots is m^2+4m+8.<br>
So now we know<br>
{{{m^5 - 11m^3-26m^2+48m+144 = (m+2)(m^2+4m+8)(m^2+am+b)}}}<br>
where the coefficients a and b in the second quadratic factor are yet to be determined.<br>
To find those coefficients, we know that<br>
{{{(m^4-2m^3-7m^2-12m+72)=(m^2+4m+8)(m^2+am+b)}}}}<br>
We can immediately see that b=9 by looking at the constant term: 72 is equal to 8 times b.<br>
And one quick way (with a little practice) to find the coefficient a is to see that the coefficient of the m^3 term, -2, comes from the two partial products (m^2)*(am) and (4m)(m^2).  So<br>
{{{-2 = a+4}}}
{{{a = -6}}}<br>
So now we know the factorization is<br>
{{{m^5 - 11m^3-26m^2+48m+144 = (m+2)(m^2+4m+8)(m^2-6m+9)}}}<br>
And finally we see that the second quadratic factor is reducible, and the final complete factorization is<br>
{{{m^5 - 11m^3-26m^2+48m+144 = (m+2)(m^2+4m+8)(m-3)^2}}}