1. **State the problem:** Simplify the expression $4a^2 + 3a + 2 \times 5a^2 - 6a + 3$. 2. **Clarify the expression:** The expression can be interpreted as $(4a^2 + 3a + 2)(5a^2 - 6a + 3)$ because multiplication is implied between the two polynomials. 3. **Recall the formula:** To multiply two polynomials, use the distributive property (FOIL for binomials, extended here): $$ (A + B + C)(D + E + F) = AD + AE + AF + BD + BE + BF + CD + CE + CF $$ 4. **Apply the distributive property:** $$ (4a^2)(5a^2) + (4a^2)(-6a) + (4a^2)(3) + (3a)(5a^2) + (3a)(-6a) + (3a)(3) + (2)(5a^2) + (2)(-6a) + (2)(3) $$ 5. **Calculate each term:** - $4a^2 \times 5a^2 = 20a^4$ - $4a^2 \times (-6a) = -24a^3$ - $4a^2 \times 3 = 12a^2$ - $3a \times 5a^2 = 15a^3$ - $3a \times (-6a) = -18a^2$ - $3a \times 3 = 9a$ - $2 \times 5a^2 = 10a^2$ - $2 \times (-6a) = -12a$ - $2 \times 3 = 6$ 6. **Combine like terms:** - $20a^4$ - $-24a^3 + 15a^3 = -9a^3$ - $12a^2 - 18a^2 + 10a^2 = 4a^2$ - $9a - 12a = -3a$ - $6$ 7. **Final simplified expression:** $$ 20a^4 - 9a^3 + 4a^2 - 3a + 6 $$