Write down and simplify the first \(4\) terms in the expansion by the binomial theorem of \(\left(1 - \tfrac{1}{2}x\right)^{10}\).

so that \[ \left(1 - \tfrac{1}{2}x\right)^{10} = 1 - 5x + \frac{45}{4} x^2 - 15x^3 + \dotsb \]

Find the coefficient of \(x^2\) in the expansion of \[ (5 + 4x)\left(1 - \tfrac{1}{2}x\right)^{10}. \]

There are only two contributing terms to the coefficient of \(x^2\).

The \(5\) from the factor \((5 + 4x)\) pairs with the \(\dfrac{45}{4}x^2\) term in the expansion of \(\left(1 - \tfrac{1}{2}x\right)^{10}\), and the \(4x\) term pairs with the \(-5x\) term. Since we have already calculated the coefficients in this expansion this is a straightforward task.

Thus, we have that the coefficient of \(x^2\) in the product is \[\begin{align*} 5 \times \frac{45}{4} + 4 \times (-5) &= \frac{225}{4} - 20 \\ &= \frac{145}{4}. \end{align*}\]