Sure, let's expand and fully simplify the expression [tex]\((x+1)(x^2 + 5x + 3)\)[/tex] step by step.
### Step 1: Distribute each term of the first binomial
We'll distribute each term in the first binomial, [tex]\(x + 1\)[/tex], to each term in the second polynomial, [tex]\(x^2 + 5x + 3\)[/tex].
[tex]\[
(x+1)(x^2 + 5x + 3) = x(x^2 + 5x + 3) + 1(x^2 + 5x + 3)
\][/tex]
### Step 2: Distribute [tex]\(x\)[/tex] to each term in [tex]\(x^2 + 5x + 3\)[/tex]
[tex]\[
x(x^2 + 5x + 3) = x \cdot x^2 + x \cdot 5x + x \cdot 3 = x^3 + 5x^2 + 3x
\][/tex]
### Step 3: Distribute [tex]\(1\)[/tex] to each term in [tex]\(x^2 + 5x + 3\)[/tex]
[tex]\[
1(x^2 + 5x + 3) = 1 \cdot x^2 + 1 \cdot 5x + 1 \cdot 3 = x^2 + 5x + 3
\][/tex]
### Step 4: Combine the results from Steps 2 and 3
Now, we combine the results of the previous steps:
[tex]\[
(x+1)(x^2 + 5x + 3) = (x^3 + 5x^2 + 3x) + (x^2 + 5x + 3)
\][/tex]
### Step 5: Combine like terms
Finally, we combine like terms to fully simplify the expression:
[tex]\[
x^3 + 5x^2 + 3x + x^2 + 5x + 3 = x^3 + (5x^2 + x^2) + (3x + 5x) + 3 = x^3 + 6x^2 + 8x + 3
\][/tex]
### Conclusion
The expanded and fully simplified form of [tex]\((x+1)(x^2 + 5x + 3)\)[/tex] is:
[tex]\[
x^3 + 6x^2 + 8x + 3
\][/tex]
