= 10.(4x² + 14)⁶.(x - 15)⁹ + 32x.(4x² + 14)⁵.(x - 15)¹⁰ → to be clearer, let: a = 4x² + 14

= 10a⁶.(x - 15)⁹ + 32x.a⁵.(x - 15)¹⁰ → to be clearer, let: b = x - 15

= 10a⁶.b⁹ + 32x.a⁵.b¹⁰ → you can factorize: a⁵

= a⁵.(10a.b⁹ + 32x.b¹⁰) → you can factorize: b⁹

= a⁵. b⁹.(10a + 32x.b) → and you can factorize: 2

= 2a⁵. b⁹.(5a + 16x.b) → recall: a = 4x² + 14

= 2.(4x² + 14).b⁹.[5.(4x² + 14) + 16x.b] → you can factorize 2 into the first (...)

= 4.(2x² + 7).b⁹.[5.(4x² + 14) + 16x.b] → you can factorize 2 into the second (...)

= 4.(2x² + 7).b⁹.[10.(2x² + 7) + 16x.b] → recall: b = x - 15

= 4.(2x² + 7).(x - 15)⁹.[10.(2x² + 7) + 16x.(x - 15)] → you can expand into the [...]

= 4.(2x² + 7).(x - 15)⁹.[20x² + 70 + 16x² - 240]

= 4.(2x² + 7).(x - 15)⁹.[36x² - 170] → you can factorize 2 into the [...]

= 4.(2x² + 7).(x - 15)⁹.2.[18x² - 85] → you simplify

= 8.(2x² + 7).(x - 15)⁹.(18x² - 85)

First we remove some of the clutter. Put (4x^2+14) = a and (x-15) = b.

Then the expression becomes  10a^6b^9 +32a^5b^10. We quickly see that 2a^5b^9 is a common factor so we have 2a^5b^9(5a+16b). Now

5a +16b = 20x^2 +70 +16x -240 = 20x^2 +16x -170 = 2(10x^2 +8x -85).

The expression then becomes 4(10x^2+16x-85)(2^5)(2x^2+7)^5(x-15)^9 =

(2^7)((10x^2+16x-85)(2x^2+7)^5(x-15)^9.

Hint:

HCF = 2(4x^2+14)^5(x−15)^9

4x^2 + 14 = a

x - 15 = b

10 * a^6 * b^9 + 32 * x * a^5 * b^10 =>

2 * a^5 * b^9 * (5 * a + 16 * x * b) =>

2 * (5a + 16bx) * a^5 * b^9 =>

2 * (5 * (4x^2 + 14) + 16 * (x - 15) * x) * (4x^2 + 14)^5 * (x - 15)^9 =>

2 * (20x^2 + 70 + 16x^2 - 240x) * (4x^2 + 14)^5 * (x - 15)^9 =>

2 * (36x^2 - 240x + 70) * 2^5 * (2x^2 + 7)^5 * (x - 15)^9 =>

2^6 * 2 * (18x^2 - 120x + 35) * (2x^2 + 7)^5 * (x - 15)^9 =>

128 * (18x^2 - 120x + 35) * (2x^2 + 7)^5 * (x - 15)^9

Let's see if 18x^2 - 120x + 35 is factorable

18x^2 - 120x + 35 = 0

x = (120 +/- sqrt(14400 - 4 * 18 * 35)) / 36

x = (120 +/- sqrt(4 * (3600 - 18 * 35))) / 36

x = (120 +/- 2 * sqrt(18 * (200 - 35))) / 36

x = (120 +/- 2 * 3 * sqrt(2 * 165)) / 36

Nope.

128 * (18x^2 - 120x + 35) * (2x^2 + 7)^5 * (x - 15)^9

10(4x²+14)⁶(x−15)⁹ + 32x(4x²+14)⁵(x−15)¹⁰

2(4x²+14)⁵(x−15)⁹ [5(4x²+14) + 16x(x−15)]

2(4x²+14)⁵(x−15)⁹ [20x² + 70 + 16x² − 240]

2(4x²+14)⁵(x−15)⁹ (36x² − 170)

2•2⁵•2(2x²+7)⁵(x−15)⁹ (18x² − 85)

128(2x²+7)⁵(x−15)⁹ (18x² − 85)

The 10 AND 32x have a gcf of 2

(4x^2+14)^5 and (4x^2+14)^6 have a gcf of (4x^2+14)^5

(x−15)^9 and (x−15)^10 have a gcf of (x−15)^9

So put the product of the gcf's outside parentheses (or brackets)

and what's left from each term inside

2(4x^2+14)^5 (x−15)^9 [5(4x^2 + 14) + 16x(x-15)]

Common term: 2 * (4x^2 + 14)^5 * (x - 15)^9 =>

10(4x^2 + 14)^6 (x - 15)^9 + 32x(4x^2 + 14)^5(x - 15)^10

= 2 * (4x^2 + 14)^5 * (x - 15)^9 (5 * (4x^2 + 14) + 16x * (x - 15))

=  2 * (4x^2 + 14)^5 * (x - 15)^9 (20x^2 + 70 + 16x^2 - 240x)

=  2 * (4x^2 + 14)^5 * (x - 15)^9 (36x^2 - 240x + 70)

= 4 (4x^2 + 14)^5 (x - 15)^9 (18x^2 - 120x + 35)

Find the common factors....

10(4x^2+14)^6(x−15)^9+32x(4x^2+14)^5(x−15)^10

(4x^2 + 14)^5 is a factor of (4x^2 + 14)^6...

2 is a factor of both 4 and 14 AND of 10 and 32

(x - 15)^9 is a factor of (x-15)^10 

It's all mechanical operation from there on.