Let's break down the multiplication of the two expressions step-by-step:
We need to find the product of \((9t - 4)\) and \((-9t - 4)\).
Step 1: Use the distributive property \(a(b + c) = ab + ac\) to multiply each term in the first polynomial by each term in the second polynomial.
[tex]\[
(9t - 4)(-9t - 4) = 9t \cdot (-9t) + 9t \cdot (-4) + (-4) \cdot (-9t) + (-4) \cdot (-4)
\][/tex]
Step 2: Perform the multiplications:
[tex]\[
9t \cdot (-9t) = -81t^2
\][/tex]
[tex]\[
9t \cdot (-4) = -36t
\][/tex]
[tex]\[
(-4) \cdot (-9t) = 36t
\][/tex]
[tex]\[
(-4) \cdot (-4) = 16
\][/tex]
Step 3: Combine all the terms:
[tex]\[
-81t^2 + (-36t) + 36t + 16
\][/tex]
Step 4: Simplify by combining like terms, namely the \(-36t\) and \(+36t\) terms:
[tex]\[
-81t^2 - 36t + 36t + 16
\][/tex]
Since \(-36t + 36t = 0\), they cancel out, leaving us with:
[tex]\[
-81t^2 + 16
\][/tex]
Thus, the product of \((9t - 4)\) and \((-9t - 4)\) is:
[tex]\[
\boxed{-81t^2 + 16}
\][/tex]
