To factor the polynomial [tex]\(8x^3 + 2x^2 - 20x - 5\)[/tex] by grouping, we need to follow these steps:
1. Group the terms in pairs to facilitate factoring by grouping:
[tex]\[
(8x^3 + 2x^2) + (-20x - 5)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- For the first group [tex]\((8x^3 + 2x^2)\)[/tex], the GCF is [tex]\(2x^2\)[/tex]:
[tex]\[
2x^2(4x + 1)
\][/tex]
- For the second group [tex]\((-20x - 5)\)[/tex], the GCF is [tex]\(-5\)[/tex]:
[tex]\[
-5(4x + 1)
\][/tex]
3. Combine the factored terms:
[tex]\[
2x^2(4x + 1) - 5(4x + 1)
\][/tex]
4. Factor out the common binomial factor [tex]\((4x + 1)\)[/tex]:
[tex]\[
(2x^2 - 5)(4x + 1)
\][/tex]
Putting it all together, the factored form of the polynomial [tex]\(8x^3 + 2x^2 - 20x - 5\)[/tex] is:
[tex]\[
(2x^2 - 5)(4x + 1)
\][/tex]
Therefore, the complete solution using the expressions provided would look like this:
[tex]\[
\begin{array}{l}
8x^3 + 2x^2 - 20x - 5 \\
(8x^3 + 2x^2) + (-20x - 5) \\
2x^2(4x + 1) - 5(4x + 1)
\end{array}
\][/tex]
The expressions to be placed are:
- For the empty space beside the first pair: [tex]\( -20 x - 5 \)[/tex]
- For the empty space in the factoring step: [tex]\( 4 x + 1 \)[/tex]
So the complete detailed solution would be:
[tex]\[
\left(8 x^3+2 x^2\right)+( -20 x - 5 ) \implies 2 x^2(4 x + 1) - 5(4 x + 1) \implies (2 x^2 - 5)(4 x+1)
\][/tex]
