Question 143267
Do you want to factor?






{{{4x^2+34x+42}}} Start with the given expression



{{{2(2x^2+17x+21)}}} Factor out the GCF {{{2}}}



Now let's focus on the inner expression {{{2x^2+17x+21}}}





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Looking at {{{2x^2+17x+21}}} we can see that the first term is {{{2x^2}}} and the last term is {{{21}}} where the coefficients are 2 and 21 respectively.


Now multiply the first coefficient 2 and the last coefficient 21 to get 42. Now what two numbers multiply to 42 and add to the  middle coefficient 17? Let's list all of the factors of 42:




Factors of 42:

1,2,3,6,7,14,21,42


-1,-2,-3,-6,-7,-14,-21,-42 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 42

1*42

2*21

3*14

6*7

(-1)*(-42)

(-2)*(-21)

(-3)*(-14)

(-6)*(-7)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">42</td><td>1+42=43</td></tr><tr><td align="center">2</td><td align="center">21</td><td>2+21=23</td></tr><tr><td align="center">3</td><td align="center">14</td><td>3+14=17</td></tr><tr><td align="center">6</td><td align="center">7</td><td>6+7=13</td></tr><tr><td align="center">-1</td><td align="center">-42</td><td>-1+(-42)=-43</td></tr><tr><td align="center">-2</td><td align="center">-21</td><td>-2+(-21)=-23</td></tr><tr><td align="center">-3</td><td align="center">-14</td><td>-3+(-14)=-17</td></tr><tr><td align="center">-6</td><td align="center">-7</td><td>-6+(-7)=-13</td></tr></table>



From this list we can see that 3 and 14 add up to 17 and multiply to 42



Now looking at the expression {{{2x^2+17x+21}}}, replace {{{17x}}} with {{{3x+14x}}} (notice {{{3x+14x}}} adds up to {{{17x}}}. So it is equivalent to {{{17x}}})


{{{2x^2+highlight(3x+14x)+21}}}



Now let's factor {{{2x^2+3x+14x+21}}} by grouping:



{{{(2x^2+3x)+(14x+21)}}} Group like terms



{{{x(2x+3)+7(2x+3)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{7}}} out of the second group



{{{(x+7)(2x+3)}}} Since we have a common term of {{{2x+3}}}, we can combine like terms


So {{{2x^2+3x+14x+21}}} factors to {{{(x+7)(2x+3)}}}



So this also means that {{{2x^2+17x+21}}} factors to {{{(x+7)(2x+3)}}} (since {{{2x^2+17x+21}}} is equivalent to {{{2x^2+3x+14x+21}}})




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So our expression goes from {{{2(2x^2+17x+21)}}} and factors further to {{{2(x+7)(2x+3)}}}



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Answer:


So {{{4x^2+34x+42}}} factors to {{{2(x+7)(2x+3)}}}