(a%5E2%2B3a%2B2)%2B6.jpeg "(a^2+3a-5)(a^2+3a+2)+6")
Expanding and Simplifying the Expression: (a^2 + 3a - 5)(a^2 + 3a + 2) + 6
This article will guide you through the steps of expanding and simplifying the given algebraic expression: (a^2 + 3a - 5)(a^2 + 3a + 2) + 6.
Step 1: Expanding the Product
We begin by expanding the product of the two binomials using the distributive property (often referred to as FOIL method). This involves multiplying each term in the first binomial by each term in the second binomial:
(a^2 + 3a - 5)(a^2 + 3a + 2) =
- a^2 * (a^2 + 3a + 2) + 3a * (a^2 + 3a + 2) - 5 * (a^2 + 3a + 2)
Now, distribute each term:
- a^4 + 3a^3 + 2a^2 + 3a^3 + 9a^2 + 6a - 5a^2 - 15a - 10
Step 2: Combining Like Terms
Next, we combine all the terms with the same powers of a:
a^4 + (3a^3 + 3a^3) + (2a^2 + 9a^2 - 5a^2) + (6a - 15a) - 10
This simplifies to:
a^4 + 6a^3 + 6a^2 - 9a - 10
Step 3: Adding the Constant Term
Finally, we add the constant term, +6, to the simplified expression:
a^4 + 6a^3 + 6a^2 - 9a - 10 + 6
Step 4: Final Simplified Expression
Combining the constant terms, we arrive at the final simplified expression:
(a^2 + 3a - 5)(a^2 + 3a + 2) + 6 = a^4 + 6a^3 + 6a^2 - 9a - 4
Therefore, the expanded and simplified form of the given expression is a^4 + 6a^3 + 6a^2 - 9a - 4.