Question

(-2 + 9square root 2)(4 + 3square root 2)

Answer 1

To simplify the expression \((-2 + 9\sqrt{2})(4 + 3\sqrt{2})\), we can use the distributive property (also known as the FOIL method for binomials):

\[
(-2 + 9\sqrt{2})(4 + 3\sqrt{2}) = (-2)(4) + (-2)(3\sqrt{2}) + (9\sqrt{2})(4) + (9\sqrt{2})(3\sqrt{2})
\]

Calculating each term:

1. \((-2)(4) = -8\)
2. \((-2)(3\sqrt{2}) = -6\sqrt{2}\)
3. \((9\sqrt{2})(4) = 36\sqrt{2}\)
4. \((9\sqrt{2})(3\sqrt{2}) = 27 \cdot 2 = 54\)

Now, combine all these results:

\[
-8 - 6\sqrt{2} + 36\sqrt{2} + 54
\]

Combining like terms:

1. For the constant terms: \(-8 + 54 = 46\)
2. For the \(\sqrt{2}\) terms: \(-6\sqrt{2} + 36\sqrt{2} = 30\sqrt{2}\)

Putting it all together, we have:

\[
46 + 30\sqrt{2}
\]

Thus, the simplified expression is:

\[
\boxed{46 + 30\sqrt{2}}
\]