```Question 416132

Looking at the expression {{{32b^2-20b-3}}}, we can see that the first coefficient is {{{32}}}, the second coefficient is {{{-20}}}, and the last term is {{{-3}}}.

Now multiply the first coefficient {{{32}}} by the last term {{{-3}}} to get {{{(32)(-3)=-96}}}.

Now the question is: what two whole numbers multiply to {{{-96}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-20}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-96}}} (the previous product).

Factors of {{{-96}}}:

1,2,3,4,6,8,12,16,24,32,48,96

-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{-96}}}.

1*(-96) = -96
2*(-48) = -96
3*(-32) = -96
4*(-24) = -96
6*(-16) = -96
8*(-12) = -96
(-1)*(96) = -96
(-2)*(48) = -96
(-3)*(32) = -96
(-4)*(24) = -96
(-6)*(16) = -96
(-8)*(12) = -96

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-20}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-96</font></td><td  align="center"><font color=black>1+(-96)=-95</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-48</font></td><td  align="center"><font color=black>2+(-48)=-46</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-32</font></td><td  align="center"><font color=black>3+(-32)=-29</font></td></tr><tr><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>-24</font></td><td  align="center"><font color=red>4+(-24)=-20</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-16</font></td><td  align="center"><font color=black>6+(-16)=-10</font></td></tr><tr><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>8+(-12)=-4</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>96</font></td><td  align="center"><font color=black>-1+96=95</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>48</font></td><td  align="center"><font color=black>-2+48=46</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>32</font></td><td  align="center"><font color=black>-3+32=29</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-4+24=20</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>-6+16=10</font></td></tr><tr><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-8+12=4</font></td></tr></table>

From the table, we can see that the two numbers {{{4}}} and {{{-24}}} add to {{{-20}}} (the middle coefficient).

So the two numbers {{{4}}} and {{{-24}}} both multiply to {{{-96}}} <font size=4><b>and</b></font> add to {{{-20}}}

Now replace the middle term {{{-20b}}} with {{{4b-24b}}}. Remember, {{{4}}} and {{{-24}}} add to {{{-20}}}. So this shows us that {{{4b-24b=-20b}}}.

{{{32b^2+highlight(4b-24b)-3}}} Replace the second term {{{-20b}}} with {{{4b-24b}}}.

{{{(32b^2+4b)+(-24b-3)}}} Group the terms into two pairs.

{{{4b(8b+1)+(-24b-3)}}} Factor out the GCF {{{4b}}} from the first group.

{{{4b(8b+1)-3(8b+1)}}} Factor out {{{3}}} 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.

{{{(4b-3)(8b+1)}}} Combine like terms. Or factor out the common term {{{8b+1}}}

===============================================================

So {{{32b^2-20b-3}}} factors to {{{(4b-3)(8b+1)}}}.

In other words, {{{32b^2-20b-3=(4b-3)(8b+1)}}}.

Note: you can check the answer by expanding {{{(4b-3)(8b+1)}}} to get {{{32b^2-20b-3}}} or by graphing the original expression and the answer (the two graphs should be identical).

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Jim```