(a-2)(a'3+2a+4)

To simplify the expression \((a - 2)(a^3 + 2a + 4)\), you can use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step multiplication:

1. \( a \cdot (a^3 + 2a + 4) = a^4 + 2a^2 + 4a \)
2. \(-2 \cdot (a^3 + 2a + 4) = -2a^3 - 4a - 8\)

Now, combine the results:

\[ a^4 + 2a^2 + 4a - 2a^3 - 4a - 8 \]

Next, combine like terms:

• The \(a^4\) term: \(a^4\)
• The \(a^3\) term: \(-2a^3\)
• The \(a^2\) term: \(2a^2\)
• The \(a\) terms: \(4a - 4a = 0\)
• The constant term: \(-8\)

So, the final simplified expression is:

\[ a^4 - 2a^3 + 2a^2 - 8 \]