Why is it that when we write the

When we use synthetic division to divide a polynomial by a linear expression (in this case, x-4), we write the result as a polynomial plus a remainder over the divisor. In your example, we have:
$3x^2+16x+37+\frac{112}{x-4}=(3x+28)+\left(\frac{149}{x-4}\right)$

The first term on the right-hand side is the quotient of the division, which is a polynomial (in this case, $3x+28$). The second term is the remainder of the division, which is a rational function (in this case, $\frac{149}{x-4}$).

Now, why doesn't the quotient have an x term in it? Well, let's think about what synthetic division is doing. It's essentially a shortcut for long division, where we're dividing a polynomial by a linear expression of the form x-a. When we perform synthetic division, we're actually dividing by x-a, not just by x. So, we're looking for a polynomial of the form Ax + B that satisfies:
$(x-a)(Ax+B)=A\cdot x^2+(B-a\cdot A)\cdot x-a\cdot B$

We want this polynomial to cancel out the first term of the dividend (in this case, $3x^2$), so we need A = 3. We also want it to cancel out the second term of the dividend (in this case, 16x), so we need $B-a\cdot A=16,\text{or}B=28$. That's why the quotient is $3x+28$.

To summarize, the reason why the quotient of a synthetic division doesn't have an x term is because we're actually dividing by a linear expression of the form x-a, not just by x. The quotient is a polynomial of degree one less than the dividend, and its coefficients are determined by the algorithm of synthetic division. The remainder is a rational function that represents the leftover terms after the division is performed.