Question

Multiply. Write your answer in simplest form.

(-4 squareroot143 - 6 squareroot91) (-squareroot143 + squareroot91)

Answer 1

We start with the expression:

\[
(-4 \sqrt{143} - 6 \sqrt{91})(-\sqrt{143} + \sqrt{91}).
\]

Let's denote \( a = \sqrt{143} \) and \( b = \sqrt{91} \) to simplify our calculations. Thus, our expression can be rewritten as:

\[
(-4a - 6b)(-a + b).
\]

Now, we will expand this expression using the distributive property (also known as the FOIL method for binomials):

\[
(-4a)(-a) + (-4a)(b) + (-6b)(-a) + (-6b)(b).
\]

Calculating each term:

1. \( (-4a)(-a) = 4a^2 \)
2. \( (-4a)(b) = -4ab \)
3. \( (-6b)(-a) = 6ab \)
4. \( (-6b)(b) = -6b^2 \)

Now, we combine the results:

\[
4a^2 - 4ab + 6ab - 6b^2.
\]

Combining like terms gives:

\[
4a^2 + 2ab - 6b^2.
\]

Next, we substitute back \( a = \sqrt{143} \) and \( b = \sqrt{91} \):

1. \( a^2 = \sqrt{143}^2 = 143 \)
2. \( b^2 = \sqrt{91}^2 = 91 \)
3. \( ab = \sqrt{143} \cdot \sqrt{91} = \sqrt{13013} \).

Now, substituting these values:

\[
4(143) + 2\sqrt{13013} - 6(91).
\]

Calculating \( 4(143) \) and \( -6(91) \):

\[
4(143) = 572,
\]
\[
-6(91) = -546.
\]

Now substitute these values back into the expression:

\[
572 + 2\sqrt{13013} - 546.
\]

Now combine the constants:

\[
572 - 546 = 26.
\]

Thus, the expression simplifies to:

\[
26 + 2\sqrt{13013}.
\]

Finally, we understand that \( 13013 \) cannot be simplified further since \( 13013 = 7 \times 1859 \) and does not contain any perfect square factors.

Therefore, the final answer in simplest form is:

\[
\boxed{26 + 2\sqrt{13013}}.
\]