Sure! Let's go through a detailed step-by-step solution for this problem.
We need to work with two functions, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], and then compute their sum and product.
1. Define the functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = x^2 + 5x + 4 \][/tex]
2. Calculate the sum [tex]\( (f + g)(x) \)[/tex]:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
Substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(f + g)(x) = (x + 3) + (x^2 + 5x + 4)
\][/tex]
Combine like terms:
[tex]\[
(f + g)(x) = x^2 + 6x + 7
\][/tex]
So, the sum of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is:
[tex]\[
(f + g)(x) = x^2 + 6x + 7
\][/tex]
3. Now, let's simplify the product [tex]\( (x + 3)(x^2 + 5x + 4) \)[/tex]:
[tex]\[
(x + 3)(x^2 + 5x + 4)
\][/tex]
To expand this product, distribute [tex]\( x + 3 \)[/tex] to each term in [tex]\( x^2 + 5x + 4 \)[/tex]:
[tex]\[
x(x^2 + 5x + 4) + 3(x^2 + 5x + 4)
\][/tex]
First, distribute [tex]\( x \)[/tex]:
[tex]\[
x \cdot x^2 = x^3, \quad x \cdot 5x = 5x^2, \quad x \cdot 4 = 4x
\][/tex]
Combine these terms:
[tex]\[
x^3 + 5x^2 + 4x
\][/tex]
Next, distribute [tex]\( 3 \)[/tex]:
[tex]\[
3 \cdot x^2 = 3x^2, \quad 3 \cdot 5x = 15x, \quad 3 \cdot 4 = 12
\][/tex]
Combine these terms:
[tex]\[
3x^2 + 15x + 12
\][/tex]
Finally, add the two sets of terms together:
[tex]\[
x^3 + 5x^2 + 4x + 3x^2 + 15x + 12
\][/tex]
Combine like terms:
[tex]\[
x^3 + 8x^2 + 19x + 12
\][/tex]
So, the sum and the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are:
[tex]\[
(f + g)(x) = x^2 + 6x + 7
\][/tex]
[tex]\[
(f \cdot g)(x) = x^3 + 8x^2 + 19x + 12
\][/tex]
These are the detailed step-by-step solutions for the given problem.
