1. The problem involves simplifying and evaluating the polynomial expression $$5x^2 + xy + 3y^2$$ given values for $x$ and $y$. 2. The formula for evaluating a polynomial is to substitute the given values of variables into the expression and perform arithmetic operations. 3. Given $x = 3$ and $y = -2$, substitute these into the expression: $$5(3)^2 + (3)(-2) + 3(-2)^2$$ 4. Calculate each term: $$5 \times 9 + (-6) + 3 \times 4 = 45 - 6 + 12$$ 5. Sum the results: $$45 - 6 + 12 = 51$$ 6. Therefore, the value of the polynomial $5x^2 + xy + 3y^2$ when $x=3$ and $y=-2$ is **51**.