Sure, let's expand the given expression step-by-step. Here is the expression we need to expand:
[tex]\[
(x+2)\left(3 x^2+5 x+10\right)
\][/tex]
We will use the distributive property (also known as the distributive law of multiplication over addition). Our goal is to multiply each term inside the parentheses by [tex]\( x \)[/tex] and then by [tex]\( 2 \)[/tex].
Firstly, distribute [tex]\( x \)[/tex] to each term inside the parentheses:
[tex]\[
x \cdot (3x^2) + x \cdot (5x) + x \cdot (10)
\][/tex]
This yields:
[tex]\[
3x^3 + 5x^2 + 10x
\][/tex]
Next, distribute [tex]\( 2 \)[/tex] to each term inside the parentheses:
[tex]\[
2 \cdot (3x^2) + 2 \cdot (5x) + 2 \cdot (10)
\][/tex]
This yields:
[tex]\[
6x^2 + 10x + 20
\][/tex]
Now, combine all these results together:
[tex]\[
3x^3 + 5x^2 + 10x + 6x^2 + 10x + 20
\][/tex]
Combine like terms:
[tex]\[
3x^3 + (5x^2 + 6x^2) + (10x + 10x) + 20
\][/tex]
[tex]\[
3x^3 + 11x^2 + 20x + 20
\][/tex]
Therefore, the expanded polynomial in standard form is:
[tex]\[
3x^3 + 11x^2 + 20x + 20
\][/tex]
