Question

question
consider the polynomial $8x^3 + 2x^2 - 20x - 5$. factor by grouping to write the polynomial in factored form.
drag each expression to the correct location on the solution. not all expressions will be used.
$8x^3 + 2x^2 - 20x - 5$
$(8x^3 + 2x^2) + (\quad)$
$2x^2 (\quad) + (\quad)(4x + 1)$
$(\quad)(4x + 1)$
$4x + 1$ $-5$ $2x^2 - 5$ $2x^2 + 5$ $5$ $4x - 1$ $-20x - 5$

Explanation:

Step1: Group the polynomial

We start with the polynomial \(8x^3 + 2x^2 - 20x - 5\). To factor by grouping, we group the first two terms and the last two terms. So, \((8x^3 + 2x^2) + (-20x - 5)\).

Step2: Factor out the GCF from each group

For the first group \((8x^3 + 2x^2)\), the greatest common factor (GCF) is \(2x^2\). Factoring that out, we get \(2x^2(4x + 1)\). For the second group \((-20x - 5)\), the GCF is \(-5\). Factoring that out, we get \(-5(4x + 1)\). So now we have \(2x^2(4x + 1) + (-5)(4x + 1)\).

Step3: Factor out the common binomial factor

Now we can see that both terms have a common binomial factor of \((4x + 1)\). Factoring that out, we get \((2x^2 - 5)(4x + 1)\).

Answer:

• First blank (inside the second group): \(-20x - 5\)
• Second blank (inside the first factored group): \(4x + 1\)
• Third blank (inside the second factored group): \(-5\)
• Fourth blank (the final factored form): \(2x^2 - 5\)

So filling in the blanks:
\((8x^3 + 2x^2) + (\boldsymbol{-20x - 5})\)
\(2x^2 (\boldsymbol{4x + 1}) + (\boldsymbol{-5})(4x + 1)\)
\((\boldsymbol{2x^2 - 5})(4x + 1)\)