### Instruction: If $a$ is a constant such that $4x^2 + 14x + a$ is the square of a binomial, then what is $a$?### Response: I notice that the expression $4x^2 + 14x + a$ has a coefficient of 4 for the quadratic term, so I wonder if it can be written as $(2x + b)^2$ for some constant $b$.
If I expand $(2x + b)^2$, I get $4x^2 + 4bx + b^2$, so I need to match the coefficients of $x$ and the constant terms.
For the coefficient of $x$, I have $4bx = 14x$, so $b = \frac{14}{4} = \frac{7}{2}$.
For the constant term, I have $b^2 = a$, so $a = (\frac{7}{2})^2 = \frac{49}{4}$.
Therefore, $a = \frac{49}{4}$.