To simplify the polynomial expression \( 23ab + (a - 2b)(11a + 4b) \), we first need to expand the second part:
\[ (a - 2b)(11a + 4b) \]
Using the distributive property (also known as the FOIL method for binomials):
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Multiply \( a \) by \( 11a \): \[ a \cdot 11a = 11a^2 \]
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Multiply \( a \) by \( 4b \): \[ a \cdot 4b = 4ab \]
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Multiply \( -2b \) by \( 11a \): \[ -2b \cdot 11a = -22ab \]
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Multiply \( -2b \) by \( 4b \): \[ -2b \cdot 4b = -8b^2 \]
Now, combining all these results together:
\[ (a - 2b)(11a + 4b) = 11a^2 + 4ab - 22ab - 8b^2 \]
Now combine like terms:
- The \( ab \) terms: \( 4ab - 22ab = -18ab \)
So, we have:
\[ 11a^2 - 18ab - 8b^2 \]
Now, add this to the \( 23ab \) from the original expression:
\[ 23ab + (11a^2 - 18ab - 8b^2) \]
Combining the \( ab \) terms:
\[ 23ab - 18ab = 5ab \]
Therefore, the entire expression simplifies to:
\[ 11a^2 + 5ab - 8b^2 \]
Thus, the simplified polynomial expression is:
\[ \boxed{11a^2 + 5ab - 8b^2} \]