Question

Find the product. $$(2 b-4)\left(b^{3}+4 b^{2}+5 b\right)$$

Short answer

The product of the given expressions is \(2b^{4} + 4b^{3} - 6b^{2} - 20b\).

Step-by-step solution

Apply the distributive law of multiplication over addition

Multiply the first term of first expression i.e., \(2b\) with all the terms of the second expression i.e., \(b^{3} + 4b^{2} + 5b\) and then repeat the same process with the second term of the first expression i.e., \(-4\). The computation details are as follows: \[2b \times b^{3} = 2b^{4}\] \[2b \times 4b^{2} = 8b^{3}\], \[2b \times 5b = 10b^{2}\], \[-4 \times b^{3} = -4b^{3}\] \[-4 \times 4b^{2} = -16b^{2}\], \[-4 \times 5b = -20b\].

Combine like terms

Combining all the terms we found after multiplication, we get: \(2b^{4} + 8b^{3} + 10b^{2} - 4b^{3} - 16b^{2} - 20b\). Simplify the equation by combining the like-terms (the terms with same degree). So the combination results in: \(2b^{4} + (8b^{3} - 4b^{3})+ (10b^{2} - 16b^{2}) - 20b = 2b^{4} + 4b^{3} - 6b^{2} - 20b\).

Writing the final answer

The final answer will be: \(2b^{4} + 4b^{3} - 6b^{2} - 20b\). This is an expression in the standard form of a polynomial.