Identify the standard algebraic identity

To expand the square of a trinomial of the form $(a + b + c)^2$, we use the algebraic identity.

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Substitute the given values into the identity

Let $a = 3x$, $b = 4y$, and $c = 5z$. Substitute these into the identity terms.

$(3x + 4y + 5z)^2 = (3x)^2 + (4y)^2 + (5z)^2 + 2(3x)(4y) + 2(4y)(5z) + 2(3x)(5z)$

Simplify the squared terms

Calculate the square of each individual term: $(3x)^2 = 9x^2$, $(4y)^2 = 16y^2$, and $(5z)^2 = 25z^2$.

$9x^2 + 16y^2 + 25z^2$

Simplify the product terms

Calculate the double products: $2(3x)(4y) = 24xy$, $2(4y)(5z) = 40yz$, and $2(3x)(5z) = 30xz$.

$24xy + 40yz + 30xz$

Combine all parts to get the final expansion

Putting the squares and product terms together yields the complete expanded expression matching Option A.

$9x^2 + 16y^2 + 25z^2 + 24xy + 40yz + 30xz$