Question

Simplify the Polynomial Expresion
23ab+(a-2b)(11a+4b)

Answer 1

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):

1. Multiply \( a \) by \( 11a \):
\[
a \cdot 11a = 11a^2
\]

2. Multiply \( a \) by \( 4b \):
\[
a \cdot 4b = 4ab
\]

3. Multiply \( -2b \) by \( 11a \):
\[
-2b \cdot 11a = -22ab
\]

4. 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:

\[
\boxed{11a^2 + 5ab - 8b^2}
\]