### Instruction: If $a^2 + b^2 = 3$ and $c^2 + d^2 = 4$, find the value of

$$(ac+bd)^2 + (bc - ad)^2.$$.### Response: I notice that this problem involves squares and sums of squares, which reminds me of the Pythagorean theorem and the identity $(x+y)^2 = x^2 + 2xy + y^2$.
I wonder if I can use these facts to simplify the expression I want to find.
I try expanding the squares and rearranging the terms as follows:

$$(ac+bd)^2 + (bc - ad)^2 = (a^2 c^2 + 2abcd + b^2 d^2) + (b^2 c^2 - 2abcd + a^2 d^2).$$
I see that the term $2abcd$ cancels out, so I get

$$(ac+bd)^2 + (bc - ad)^2 = a^2 c^2 + b^2 d^2 + b^2 c^2 + a^2 d^2.$$
Now I notice that this expression is a sum of four products of squares, and I wonder if I can use the given equations $a^2 + b^2 = 3$ and $c^2 + d^2 = 4$ to simplify them.
I recall that $(a^2 + b^2)(c^2 + d^2) = a^2 c^2 + b^2 c^2 + a^2 d^2 + b^2 d^2$, which is exactly what I have.