Questions: Find the product. (x+5)(x^2+3x+9) (x+5)(x^2+3x+9)= (Simplify your answer.)

Transcript text: 8. Find the product. \[ \begin{array}{l} \quad(x+5)\left(x^{2}+3 x+9\right) \\ (x+5)\left(x^{2}+3 x+9\right)=\square \text { (Simplify your answer.) } \end{array} \]

Solution

Solution Steps

Step 1: Distribute the Terms

We start with the expression \((x + 5)(x^2 + 3x + 9)\). We will distribute each term in the first polynomial to each term in the second polynomial.

Step 2: Multiply Each Term

Calculating the products: \[ x \cdot x^2 = x^3 \] \[ x \cdot 3x = 3x^2 \] \[ x \cdot 9 = 9x \] \[ 5 \cdot x^2 = 5x^2 \] \[ 5 \cdot 3x = 15x \] \[ 5 \cdot 9 = 45 \]

Step 3: Combine Like Terms

Now, we combine all the terms obtained from the distribution: \[ x^3 + 3x^2 + 9x + 5x^2 + 15x + 45 \] Combining the like terms: \[ x^3 + (3x^2 + 5x^2) + (9x + 15x) + 45 = x^3 + 8x^2 + 24x + 45 \]

Step 4: Final Expression

The simplified expression for the product \((x + 5)(x^2 + 3x + 9)\) is: \[ x^3 + 8x^2 + 24x + 45 \]