1. The problem is to fully simplify the expression $5a^3 f^4 \times 3a^2 f^3$. 2. Start by multiplying the coefficients (numbers) together: $5 \times 3 = 15$. 3. Next, multiply the powers of $a$. When multiplying like bases, add the exponents: $a^3 \times a^2 = a^{3+2} = a^5$. 4. Then, multiply the powers of $f$ similarly: $f^4 \times f^3 = f^{4+3} = f^7$. 5. Combine all parts to get the simplified expression: $$15a^5 f^7$$. Therefore, the fully simplified form of $5a^3 f^4 \times 3a^2 f^3$ is $$15a^5 f^7$$.