```Question 200463
I'm assuming that you want to factor these expressions.

# 1

{{{-15(2x^2-x-6)}}} Factor out the GCF {{{-15}}}

Now let's focus on the inner expression {{{2x^2-x-6}}}

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Looking at {{{2x^2-x-6}}} we can see that the first term is {{{2x^2}}} and the last term is {{{-6}}} where the coefficients are 2 and -6 respectively.

Now multiply the first coefficient 2 and the last coefficient -6 to get -12. Now what two numbers multiply to -12 and add to the  middle coefficient -1? Let's list all of the factors of -12:

Factors of -12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -12

(1)*(-12)

(2)*(-6)

(3)*(-4)

(-1)*(12)

(-2)*(6)

(-3)*(4)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-12</td><td>1+(-12)=-11</td></tr><tr><td align="center">2</td><td align="center">-6</td><td>2+(-6)=-4</td></tr><tr><td align="center">3</td><td align="center">-4</td><td>3+(-4)=-1</td></tr><tr><td align="center">-1</td><td align="center">12</td><td>-1+12=11</td></tr><tr><td align="center">-2</td><td align="center">6</td><td>-2+6=4</td></tr><tr><td align="center">-3</td><td align="center">4</td><td>-3+4=1</td></tr></table>

From this list we can see that 3 and -4 add up to -1 and multiply to -12

Now looking at the expression {{{2x^2-x-6}}}, replace {{{-x}}} with {{{3x+-4x}}} (notice {{{3x+-4x}}} adds up to {{{-x}}}. So it is equivalent to {{{-x}}})

{{{2x^2+highlight(3x+-4x)+-6}}}

Now let's factor {{{2x^2+3x-4x-6}}} by grouping:

{{{(2x^2+3x)+(-4x-6)}}} Group like terms

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

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

So {{{2x^2+3x-4x-6}}} factors to {{{(x-2)(2x+3)}}}

So this also means that {{{2x^2-x-6}}} factors to {{{(x-2)(2x+3)}}} (since {{{2x^2-x-6}}} is equivalent to {{{2x^2+3x-4x-6}}})

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So our expression goes from {{{-15(2x^2-x-6)}}} and factors further to {{{-15(x-2)(2x+3)}}}

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So {{{-30x^2+15x+90}}} completely factors to {{{-15(x-2)(2x+3)}}}

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# 2

Looking at {{{8x^2+10x-3}}} we can see that the first term is {{{8x^2}}} and the last term is {{{-3}}} where the coefficients are 8 and -3 respectively.

Now multiply the first coefficient 8 and the last coefficient -3 to get -24. Now what two numbers multiply to -24 and add to the  middle coefficient 10? Let's list all of the factors of -24:

Factors of -24:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24

(1)*(-24)

(2)*(-12)

(3)*(-8)

(4)*(-6)

(-1)*(24)

(-2)*(12)

(-3)*(8)

(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 10

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-24</td><td>1+(-24)=-23</td></tr><tr><td align="center">2</td><td align="center">-12</td><td>2+(-12)=-10</td></tr><tr><td align="center">3</td><td align="center">-8</td><td>3+(-8)=-5</td></tr><tr><td align="center">4</td><td align="center">-6</td><td>4+(-6)=-2</td></tr><tr><td align="center">-1</td><td align="center">24</td><td>-1+24=23</td></tr><tr><td align="center">-2</td><td align="center">12</td><td>-2+12=10</td></tr><tr><td align="center">-3</td><td align="center">8</td><td>-3+8=5</td></tr><tr><td align="center">-4</td><td align="center">6</td><td>-4+6=2</td></tr></table>

From this list we can see that -2 and 12 add up to 10 and multiply to -24

Now looking at the expression {{{8x^2+10x-3}}}, replace {{{10x}}} with {{{-2x+12x}}} (notice {{{-2x+12x}}} adds up to {{{10x}}}. So it is equivalent to {{{10x}}})

{{{8x^2+highlight(-2x+12x)+-3}}}

Now let's factor {{{8x^2-2x+12x-3}}} by grouping:

{{{(8x^2-2x)+(12x-3)}}} Group like terms

{{{2x(4x-1)+3(4x-1)}}} Factor out the GCF of {{{2x}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group

{{{(2x+3)(4x-1)}}} Since we have a common term of {{{4x-1}}}, we can combine like terms

So {{{8x^2-2x+12x-3}}} factors to {{{(2x+3)(4x-1)}}}

So this also means that {{{8x^2+10x-3}}} factors to {{{(2x+3)(4x-1)}}} (since {{{8x^2+10x-3}}} is equivalent to {{{8x^2-2x+12x-3}}})

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