Sure, let's multiply the binomial [tex]\((x+6)\)[/tex] by the trinomial [tex]\((x^2 + 3x + 5)\)[/tex] using the indicated methods.
### (a) Using the Distributive Property:
To use the Distributive Property, we will multiply each term of the binomial [tex]\((x+6)\)[/tex] by each term of the trinomial [tex]\((x^2 + 3x + 5)\)[/tex] and then combine like terms.
First, distribute [tex]\(x\)[/tex] to each term in the trinomial:
[tex]\[ x \cdot x^2 = x^3 \][/tex]
[tex]\[ x \cdot 3x = 3x^2 \][/tex]
[tex]\[ x \cdot 5 = 5x \][/tex]
Next, distribute [tex]\(6\)[/tex] to each term in the trinomial:
[tex]\[ 6 \cdot x^2 = 6x^2 \][/tex]
[tex]\[ 6 \cdot 3x = 18x \][/tex]
[tex]\[ 6 \cdot 5 = 30 \][/tex]
Now, combine all these terms together:
[tex]\[ x^3 + 3x^2 + 5x + 6x^2 + 18x + 30 \][/tex]
Finally, combine like terms:
[tex]\[ x^3 + (3x^2 + 6x^2) + (5x + 18x) + 30 \][/tex]
[tex]\[ x^3 + 9x^2 + 23x + 30 \][/tex]
Thus, using the Distributive Property, the result is:
[tex]\[ x^3 + 9x^2 + 23x + 30 \][/tex]
### (b) Using the Vertical Method:
The Vertical Method involves writing the binomial and the trinomial vertically and multiplying in a manner analogous to traditional multiplication.
Write the expressions vertically aligning the terms by degrees of [tex]\(x\)[/tex]:
[tex]\[
\begin{array}{r}
x^2 + 3x + 5 \\
\times (x + 6) \\
\hline
\end{array}
\][/tex]
First, multiply [tex]\(6\)[/tex] by each term of the trinomial:
[tex]\[
\begin{array}{r}
6 \cdot x^2 = 6x^2 \\
6 \cdot 3x = 18x \\
6 \cdot 5 = 30 \\
\hline
6x^2 + 18x + 30 \\
\end{array}
\][/tex]
Next, multiply [tex]\(x\)[/tex] by each term of the trinomial, but we need to remember to shift one place to the left, similar to traditional multiplication:
[tex]\[
\begin{array}{r}
x \cdot x^2 = x^3 \\
x \cdot 3x = 3x^2 \\
x \cdot 5 = 5x \\
\hline
x^3 + 3x^2 + 5x \\
\end{array}
\][/tex]
Now, add the results from these two multiplications:
[tex]\[
\begin{array}{r}
x^3 + 3x^2 + 5x \\
+ \ \ \ \ 6x^2 + 18x + 30 \\
\hline
x^3 + 9x^2 + 23x + 30 \\
\end{array}
\][/tex]
So, using the Vertical Method, the result is also:
[tex]\[ x^3 + 9x^2 + 23x + 30 \][/tex]
### Summary
Both methods give the same final result:
[tex]\[ x^3 + 9x^2 + 23x + 30 \][/tex]
