Help with polynomials

Here's another way to determine t  with polynomial long division

                                 4x^2  -  6x  + 10

5x^2 - 6x + 7     [     20x^4 -  54x^3  + 114x^2  - 102x  + 70  ] 

                                20x^4 - 24x^3  +    28x^2

                                ________________________________

                                             - 30x^3 +  86x^2  - 102x   

                                            - 30x^3  + 36x^2  -    42x

                                            _____________________  

                                                            50x^2  -  60x  + 70

                                                            50x^2 -  60x   + 70

                                                           ___________________

So...it appears that  t =  -6

I hope you meant solve for the variable t, because this took me too long for me to get mad if that's not what you meant

(5x2−6x+7)(4x2+tx+10)=20x4−54x3+114x2−102x+70

Step 1: Add -70 to both sides.

5tx3+20x4−6tx2−24x3+7tx+78x2−60x+70+−70=20x4−54x3+114x2−102x+70+−70

5tx3+20x4−6tx2−24x3+7tx+78x2−60x=20x4−54x3+114x2−102x

Step 2: Add -78x^2 to both sides.

5tx3+20x4−6tx2−24x3+7tx+78x2−60x+−78x2=20x4−54x3+114x2−102x+−78x2

5tx3+20x4−6tx2−24x3+7tx−60x=20x4−54x3+36x2−102x

Step 3: Add -20x^4 to both sides.

5tx3+20x4−6tx2−24x3+7tx−60x+−20x4=20x4−54x3+36x2−102x+−20x4

5tx3−6tx2−24x3+7tx−60x=−54x3+36x2−102x

Step 4: Add 60x to both sides.

5tx3−6tx2−24x3+7tx−60x+60x=−54x3+36x2−102x+60x

5tx3−6tx2−24x3+7tx=−54x3+36x2−42x

Step 5: Add 24x^3 to both sides.

5tx3−6tx2−24x3+7tx+24x3=−54x3+36x2−42x+24x3

5tx3−6tx2+7tx=−30x3+36x2−42x

Step 6: Factor out variable t.

t(5x3−6x2+7x)=−30x3+36x2−42x

Step 7: Divide both sides by 5x^3-6x^2+7x.

t(5x3−6x2+7x)5x3−6x2+7x=−30x3+36x2−42x5x3−6x2+7x

t=−30x2+36x−425x2−6x+7

Answer:

t=−30x2+36x−425x2−6x+7

Another way is simply to set x = 1, to get:

6*(t + 14) = 48

t + 14 = 8

t = 8 - 14 → -6

(in fact you can set x to be pretty much anything you like, apart from zero, and you'll get the same result!)

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