Let's find the product step-by-step:
First, given the two expressions:
[tex]\[ (9t - 4) \][/tex]
and
[tex]\[ (-9t - 4) \][/tex]
We need to calculate the product of these two expressions:
[tex]\[ (9t - 4)(-9t - 4) \][/tex]
To multiply these expressions, we apply the distributive property (also known as the FOIL method for binomials):
1. First terms:
[tex]\[ 9t \times (-9t) = -81t^2 \][/tex]
2. Outer terms:
[tex]\[ 9t \times (-4) = -36t \][/tex]
3. Inner terms:
[tex]\[ -4 \times (-9t) = 36t \][/tex]
4. Last terms:
[tex]\[ -4 \times (-4) = 16 \][/tex]
Now, let's combine all these terms:
[tex]\[ -81t^2 - 36t + 36t + 16 \][/tex]
Notice that the middle terms \(-36t\) and \(36t\) cancel each other out:
[tex]\[ -81t^2 + 16 \][/tex]
Therefore, the product of \((9t - 4)(-9t - 4)\) is:
[tex]\[ -81t^2 + 16 \][/tex]
Among the given options, the correct answer is:
[tex]\[ -81t^2 + 16 \][/tex]
