Step 1 :Given the algebraic expression $3(x+y)^{2}+5(x+y)$
Step 2 :We can simplify it by using the distributive property of multiplication over addition. This means we can multiply each term inside the parentheses by the term outside the parentheses.
Step 3 :Applying the distributive property, we get $5x + 5y + 3(x^{2} + 2xy + y^{2})$
Step 4 :Further simplifying, we get $5x + 5y + 3x^{2} + 6xy + 3y^{2}$
Step 5 :Rearranging the terms, we get $3x^{2} + 6xy + 5x + 3y^{2} + 5y$
Step 6 :\(\boxed{3x^{2} + 6xy + 5x + 3y^{2} + 5y}\) is the simplified form of the given expression.