Decide whether the following statement is true or false. A polynomial function of degree 4 with real coefficients could have $-3,1+i, 1-i$, and $-3+7 i$ as its zeros.

Step 1 ：Decide whether the following statement is true or false: A polynomial function of degree 4 with real coefficients could have $-3,1+i, 1-i$, and $-3+7 i$ as its zeros.

Step 2 ：A polynomial function of degree 4 with real coefficients can have complex roots, but they must come in conjugate pairs. This is because the coefficients of the polynomial are real numbers, and when you multiply out \((x - (a + bi))(x - (a - bi))\), the imaginary parts cancel out, leaving only real coefficients.

Step 3 ：In this case, $1+i$ and $1-i$ are conjugate pairs, but $-3$ and $-3+7i$ are not. Therefore, the statement is false.

Step 4 ：Final Answer: The statement is \(\boxed{false}\).