Problem 144 - Algebra and Trigonometry

Question

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right)$$

Step-by-Step Solution

Answer

After the steps of distribution and combination of like terms, the final simplified result is \(2x^3 + 11x^2 + 22x + 15\).

Step 1 - Distribute the terms

First off, distribute each term outside of the parentheses with every term inside the parentheses. For the first part, this means multiplying 2x to every term of the bracket \(x^{2} + 4x + 5\), giving \(2x^3 + 8x^2 + 10x\). Likewise, distribute the 3 to the same parentheses, resulting in \(3x^2 + 12x + 15\). The expression now reads \(2x^3 + 8x^2 + 10x + 3x^2 + 12x + 15\).

Step 2 - Combine like terms

The like terms are identified as those with equal powers of x, these can all be placed together in the equation. This evaluation leads to: \(2x^3 + (8x^2 + 3x^2) + (10x + 12x) + 15\). Initial simplification brings this to \(2x^3 + 11x^2 + 22x + 15\). In this stage, the question asks for the simplification presented in descending powers of x, therefore, the solution is given as the answer.