1. \( 2 x^{4} y^{5} \cdot 7 x^{3} y^{7} \)

To simplify the expression \( 2 x^{4} y^{5} \cdot 7 x^{3} y^{7} \), you first multiply the coefficients (the numbers in front). So, \( 2 \cdot 7 = 14 \). Next, you combine the like terms for \( x \) and \( y \) by adding their exponents: \( x^{4+3} = x^{7} \) and \( y^{5+7} = y^{12} \). Therefore, the simplified form is \( 14 x^{7} y^{12} \). In polynomial multiplication, always remember to multiply the coefficients and then follow the rules of exponents for combining like bases. A common mistake is forgetting to add the exponents for the same bases or miscalculating the coefficients. So take your time, and double-check your math to avoid these snags!