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Factoring and Simplifying the Expression: (p^2 + p - 6)(p^2 - 6)
This expression involves the multiplication of two quadratic expressions. To simplify it, we can use the following steps:
Step 1: Factor each quadratic expression
- (p^2 + p - 6) can be factored as (p + 3)(p - 2). To factor this, we need to find two numbers that multiply to -6 and add up to 1 (the coefficient of the middle term). The numbers 3 and -2 satisfy these conditions.
- (p^2 - 6) cannot be factored further as it's a difference of squares.
Step 2: Multiply the factored expressions
Now we have: (p + 3)(p - 2)(p^2 - 6)
Step 3: Expanding the expression (optional)
While the expression is simplified by factoring, we can further expand it by multiplying the terms:
(p + 3)(p - 2)(p^2 - 6) = (p^3 - 2p^2 + 3p^2 - 6p)(p^2 - 6) = (p^3 + p^2 - 6p)(p^2 - 6) = p^5 - 6p^3 + p^4 - 6p^2 - 6p^3 + 36p = p^5 + p^4 - 12p^3 - 6p^2 + 36p
Therefore, the simplified form of (p^2 + p - 6)(p^2 - 6) is (p + 3)(p - 2)(p^2 - 6), or in expanded form, p^5 + p^4 - 12p^3 - 6p^2 + 36p.