etermine the product and simplify: \[ a\left(x^{2}+3 y\right)+a x+4 a y \]

To determine the product and simplify the expression \( a\left(x^{2}+3 y\right)+a x+4 a y \), we start by distributing \( a \): \[ a\left(x^{2}+3 y\right) = a x^{2} + 3 a y \] Now we can rewrite the entire expression by substituting this back in: \[ a x^{2} + 3 a y + a x + 4 a y \] Combine like terms. The terms with \( y \) can be combined: \[ 3 a y + 4 a y = 7 a y \] Thus, the simplified expression becomes: \[ a x^{2} + a x + 7 a y \] This is the final simplified form of the expression.