Let's solve the given problem step-by-step.
We are given the equation [tex]\( x + y = k \)[/tex] and need to find the value of the expression [tex]\( 3x^2 + 6xy + 3y^2 \)[/tex].
Firstly, let's express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( k \)[/tex] from the equation [tex]\( x + y = k \)[/tex]:
[tex]\[ y = k - x \][/tex]
Now, substitute [tex]\( y = k - x \)[/tex] into the expression [tex]\( 3x^2 + 6xy + 3y^2 \)[/tex]:
[tex]\[ 3x^2 + 6x(k - x) + 3(k - x)^2 \][/tex]
Next, expand the expression:
[tex]\[ 3x^2 + 6xk - 6x^2 + 3(k^2 - 2kx + x^2) \][/tex]
Simplify by distributing and combining like terms:
[tex]\[ 3x^2 + 6xk - 6x^2 + 3k^2 - 6kx + 3x^2 \][/tex]
Combine like terms:
[tex]\[ (3x^2 - 6x^2 + 3x^2) + (6xk - 6xk) + 3k^2 \][/tex]
This simplifies to:
[tex]\[ 3k^2 \][/tex]
Therefore, the value of the expression [tex]\( 3x^2 + 6xy + 3y^2 \)[/tex] is [tex]\( \boxed{3k^2} \)[/tex].
So, the correct answer is:
(d) [tex]\( 3k^2 \)[/tex]
