Find the coefficient of the $x^2$ term in the expansion of the product $$(2x^2 +3x +4)(5x^2 +6x +7).$$
To find the coefficient of the $x^2$ term, I need to multiply the terms that will give me an $x^2$ term when I add their exponents. There are three ways to do this: multiply the $2x^2$ term by the constant term $7$, multiply the $3x$ term by the $6x$ term, or multiply the constant term $4$ by the $5x^2$ term. I can ignore the rest of the terms, since they will either give me a higher or lower power of $x$. So, the $x^2$ term in the expansion will be $$2x^2 \cdot 7 + 3x \cdot 6x + 4 \cdot 5x^2 = 14x^2 + 18x^2 + 20x^2 = 52x^2.$$ Therefore, the coefficient of the $x^2$ term is 52. 