Question

Find the product. Simplify your answer.
(4f+1)(f2+3f+4)

Answer 1

To find the product of \((4f + 1)(f^2 + 3f + 4)\), we can use the distributive property (also known as the FOIL method for binomials). We'll distribute each term in the first expression to each term in the second expression:

\[
(4f + 1)(f^2 + 3f + 4) = 4f \cdot (f^2 + 3f + 4) + 1 \cdot (f^2 + 3f + 4)
\]

Now we will distribute \(4f\):

\[
4f \cdot f^2 = 4f^3
\]
\[
4f \cdot 3f = 12f^2
\]
\[
4f \cdot 4 = 16f
\]

Next, we distribute \(1\):

\[
1 \cdot f^2 = f^2
\]
\[
1 \cdot 3f = 3f
\]
\[
1 \cdot 4 = 4
\]

Now we can combine all these terms:

\[
4f^3 + 12f^2 + 16f + f^2 + 3f + 4
\]

Now, combine like terms:

- The \(f^3\) term: \(4f^3\)
- The \(f^2\) terms: \(12f^2 + f^2 = 13f^2\)
- The \(f\) terms: \(16f + 3f = 19f\)
- The constant term: \(4\)

Putting it all together, we get:

\[
\boxed{4f^3 + 13f^2 + 19f + 4}
\]