1. Let's start by understanding the problem: you have an expression after expansion: $$144a - 48a + 2520d - 1128d = 0$$ and you want to verify if this leads to $S_{30} = 870$. 2. First, simplify the expression by combining like terms: $$144a - 48a = (144 - 48)a = 96a$$ $$2520d - 1128d = (2520 - 1128)d = 1392d$$ So the equation becomes: $$96a + 1392d = 0$$ 3. To isolate $a$, subtract $1392d$ from both sides: $$96a = -1392d$$ Now divide both sides by 96: $$a = \frac{-1392d}{96}$$ Simplify the fraction by canceling common factors: $$a = \frac{-\cancel{1392} \times 1d}{\cancel{96} \times 1} = -14.5d$$ 4. You mentioned getting positive $14.5d$, but the calculation shows $a = -14.5d$. Please check the sign carefully. 5. Next, you wrote $15[29d + 29d]$. Simplify inside the brackets: $$29d + 29d = 58d$$ Multiply by 15: $$15 \times 58d = 870d$$ 6. If $S_{30} = 870$, and assuming $S_n$ is the sum of an arithmetic sequence, the formula is: $$S_n = \frac{n}{2} [2a + (n-1)d]$$ For $n=30$: $$S_{30} = 15 [2a + 29d]$$ 7. Substitute $a = -14.5d$ into the sum formula: $$S_{30} = 15 [2(-14.5d) + 29d] = 15 [-29d + 29d] = 15 \times 0 = 0$$ 8. This contradicts $S_{30} = 870$. So either the value of $a$ or $d$ or the sign is incorrect. 9. If you want $S_{30} = 870$, solve for $d$: $$870 = 15 [2a + 29d]$$ Substitute $a = -14.5d$: $$870 = 15 [2(-14.5d) + 29d] = 15 [-29d + 29d] = 15 \times 0 = 0$$ Again zero, so no solution unless $a$ or $d$ changes. 10. Please recheck your initial values or signs to ensure consistency. Final answer: Based on your expansion and substitution, $S_{30}$ cannot be 870 with $a = -14.5d$.