343=73 and 8=23 so it would appear that 2x+7 could be a factor, since all the coefficients are positive. To test this, use synthetic division:
If 2x+7 is a factor, then 2x+7=0, x=-7/2 is a zero.
Try x=-7/2 as the divisor:
-7/2| 8 84 294 343
8 -28 -196 | -343
8 56 98 | 0 = 8x2+56x+98 with no remainder, confirming that 2x+7 is a factor.
Let's rewrite the original equation: (8x3+84x2+294x+343)/2=4x3+42x2+147x+343/2. The zeroes of this polynomial are the same those in the original polynomial. We can similarly halve the quotient: 4x2+28x+49.
2x+7 is a factor of this polynomial. So now we have to have to factorise the quadratic 4x2+28x+49=(2x+7)2.
So the original polynomial is (2x+7)3.