Certainly! Let's find the product of the expression [tex]\((x + 9)(4x^2 + 5x + 6)\)[/tex] step by step.
To multiply these polynomials, we use the distributive property, which states that [tex]\(a(b + c) = ab + ac\)[/tex]. For our expression, this means we will multiply each term in the first polynomial by each term in the second polynomial.
First, let's distribute [tex]\(x\)[/tex]:
1. [tex]\(x \cdot 4x^2 = 4x^3\)[/tex]
2. [tex]\(x \cdot 5x = 5x^2\)[/tex]
3. [tex]\(x \cdot 6 = 6x\)[/tex]
Next, let's distribute [tex]\(9\)[/tex]:
4. [tex]\(9 \cdot 4x^2 = 36x^2\)[/tex]
5. [tex]\(9 \cdot 5x = 45x\)[/tex]
6. [tex]\(9 \cdot 6 = 54\)[/tex]
Now, let's combine all these terms:
[tex]\[ 4x^3 + 5x^2 + 6x + 36x^2 + 45x + 54 \][/tex]
Next, we combine like terms. Group the terms with [tex]\(x^2\)[/tex] together and [tex]\(x\)[/tex] together:
1. [tex]\(4x^3\)[/tex]
2. [tex]\(5x^2 + 36x^2 = 41x^2\)[/tex]
3. [tex]\(6x + 45x = 51x\)[/tex]
4. [tex]\(54\)[/tex]
So, the simplified expression is:
[tex]\[ 4x^3 + 41x^2 + 51x + 54 \][/tex]
Therefore, the product of [tex]\((x + 9)(4x^2 + 5x + 6)\)[/tex] is:
[tex]\[ \boxed{4x^3 + 41x^2 + 51x + 54} \][/tex]
