Identify terms in the formula

Map the components of the expression $(5a + 6b + 8c)^2$ to the identity $(x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$, where $x=5a$, $y=6b$, and $z=8c$.

$x=5a, y=6b, z=8c$

Calculate square terms

Compute the squares of individual terms: $(5a)^2 = 25a^2$, $(6b)^2 = 36b^2$, and $(8c)^2 = 64c^2$.

$25a^2 + 36b^2 + 64c^2$

Calculate product terms

Compute twice the cross-products: $2(5a)(6b) = 60ab$, $2(6b)(8c) = 96bc$, and $2(5a)(8c) = 80ac$.

$60ab + 96bc + 80ac$

Combine result

Summing all the parts gives the final expanded form $25a^2 + 36b^2 + 64c^2 + 60ab + 96bc + 80ac$.

$25a^2 + 36b^2 + 64c^2 + 60ab + 96bc + 80ac$