### Instruction: Find the constant $t$ such that \[(5x^2 - 6x + 7)(4x^2 +tx + 10) = 20x^4 -54x^3 +114x^2 -102x +70.\].### Response: I see that this problem is about polynomial multiplication and comparing coefficients of equal powers of $x$.
To multiply two polynomials, I can use the distributive property and combine like terms.
I will start by multiplying the first terms of each factor, which gives me $20x^4$.
This matches the first term of the product on the right, so that's good.
Next, I will multiply the first term of the first factor by the second term of the second factor, which gives me $5tx^3$.
Then, I will multiply the second term of the first factor by the first term of the second factor, which gives me $-24x^3$.
I will add these two terms and get $(5t - 24)x^3$.
This has to match the coefficient of $x^3$ on the right, which is $-54$.
So I have the equation $5t - 24 = -54$.
I can solve this by adding $24$ to both sides and dividing by $5$.
I get $t = -6$.
This is the constant I need to find.