Multiply. \( \begin{array}{l}\left(a^{2}+a-1\right)\left(a^{2}+6 a-8\right) \\ \\ \left(a^{2}+a-1\right)\left(a^{2}+6 a-8\right)=\square \\ \text { (Simplify your answer.) }\end{array} \)
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Beyond the Answer
To multiply \( \left(a^{2}+a-1\right)\left(a^{2}+6 a-8\right) \), we will use the distributive property (also known as the FOIL method for binomials). 1. Distribute each term in the first polynomial by each term in the second polynomial: \[ (a^2)(a^2) + (a^2)(6a) + (a^2)(-8) + (a)(a^2) + (a)(6a) + (a)(-8) + (-1)(a^2) + (-1)(6a) + (-1)(-8) \] 2. We get: \[ a^4 + 6a^3 - 8a^2 + a^3 + 6a^2 - 8a - a^2 - 6a + 8 \] 3. Combine like terms: - \( a^4 \) - \( 6a^3 + a^3 = 7a^3 \) - \( -8a^2 + 6a^2 - a^2 = -3a^2 \) - \( -8a - 6a = -14a \) - \( +8 \) Thus, the simplified result is: \[ \boxed{a^4 + 7a^3 - 3a^2 - 14a + 8} \]
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