To simplify the expression \((5.3a^2 - 0.4a + 9.1) - (4.2 + 3.8a^2 - 6.7a)\), we will first distribute the negative sign across the second set of parentheses.
Starting with the original expression:
\[ (5.3a^2 - 0.4a + 9.1) - (4.2 + 3.8a^2 - 6.7a) \]
We can rewrite it as:
\[ 5.3a^2 - 0.4a + 9.1 - 4.2 - 3.8a^2 + 6.7a \]
Now, we'll group like terms together:
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For \(a^2\) terms:
\(5.3a^2 - 3.8a^2 = (5.3 - 3.8)a^2 = 1.5a^2\)
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For \(a\) terms:
\(-0.4a + 6.7a = (6.7 - 0.4)a = 6.3a\)
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For constant terms:
\(9.1 - 4.2 = 4.9\)
Now, combining all of these results together, we have:
\[ 1.5a^2 + 6.3a + 4.9 \]
Thus, the expression equivalent to \((5.3a^2 - 0.4a + 9.1) - (4.2 + 3.8a^2 - 6.7a)\) is:
\[ \boxed{1.5a^2 + 6.3a + 4.9} \]
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