Consider the polynomial [tex]8x^3 + 2x^2 - 20x - 5[/tex]. 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.

[tex]\[
\begin{array}{l}
8x^3 + 2x^2 - 20x - 5 \\
\left(8x^3 + 2x^2\right) + \left(-20x - 5\right) \\
2x^2(4x + 1) - 5(4x + 1)
\end{array}
\][/tex]

Expressions:
- [tex]4x + 1[/tex]
- [tex]-5[/tex]
- [tex]2x^2 - 5[/tex]
- [tex]2x^2 + 5[/tex]
- [tex]5[/tex]
- [tex]4x - 1[/tex]
- [tex]-20x - 5[/tex]

Drag the correct expressions to complete the factorization:

[tex]\[
2x^2(4x + 1) - 5(4x + 1)
\][/tex]

Final factored form:

[tex]\[
(2x^2 - 5)(4x + 1)
\][/tex]



Answer :

To factor the polynomial [tex]\( 8x^3 + 2x^2 - 20x - 5 \)[/tex] by grouping, we can follow these steps:

1. Group Terms: First, we can group the terms in pairs to facilitate factorization by grouping:
[tex]\[ 8x^3 + 2x^2 - 20x - 5 = (8x^3 + 2x^2) + (-20x - 5) \][/tex]

2. Factor Each Group: Next, we factor out the greatest common factor (GCF) from each group:
[tex]\[ (8x^3 + 2x^2) + (-20x - 5) = 2x^2(4x + 1) - 5(4x + 1) \][/tex]

3. Factor Out the Common Binomial Factor: Notice the common binomial factor [tex]\((4x + 1)\)[/tex] in both groups. We can factor this out:
[tex]\[ 2x^2(4x + 1) - 5(4x + 1) = (4x + 1)(2x^2 - 5) \][/tex]

4. Verify the Factorization: To ensure the factorization is correct, we can expand the factored form and check if it matches the original polynomial:
[tex]\[ (4x + 1)(2x^2 - 5) = 4x(2x^2) - 4x(5) + 1(2x^2) - 1(5) = 8x^3 - 20x + 2x^2 - 5 \][/tex]

This matches the original polynomial.

Thus, the factored form of [tex]\( 8x^3 + 2x^2 - 20x - 5 \)[/tex] by grouping is:
[tex]\[ (4x + 1)(2x^2 - 5) \][/tex]

Let’s map each expression to its correct place in the solution:

[tex]\[ 8 x^3+2 x^2-20 x-5 = (8 x^3+2x^2) + (-20x - 5) \][/tex]

[tex]\[ (8 x^3+2x^2)+(-20x-5) = 2 x^2(4 x + 1) - 5(4 x + 1) \][/tex]

[tex]\(\boxed{4x+1}\)[/tex]
[tex]\(\boxed{-5}\)[/tex]

Thus:
[tex]\[ 8 x^3 + 2 x^2 - 20 x - 5 = (4x + 1)(2x^2 - 5) \][/tex]