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]
