In the expansion of `(5x-3) (x + 2)^(2)`, find :
coefficients of `x^(2)` and x

Answer

To solve the problem of finding the coefficients of \(x^2\) and \(x\) in the expansion of \((5x - 3)(x + 2)^2\), we can follow these steps:

### Step 1: Expand \((x + 2)^2\)
Using the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\), we can expand \((x + 2)^2\):

\[ (x + 2)^2 = x^2 + 2 \cdot x \cdot 2 + 2^2 = x^2 + 4x + 4 \]

### Step 2: Substitute the expansion back into the expression
Now, substitute the expansion of \((x + 2)^2\) back into the original expression:

\[ (5x - 3)(x^2 + 4x + 4) \]

### Step 3: Distribute \((5x - 3)\) across \((x^2 + 4x + 4)\)
Now, we will distribute \(5x - 3\) to each term in the polynomial:

\[ = 5x(x^2) + 5x(4x) + 5x(4) - 3(x^2) - 3(4x) - 3(4) \]

Calculating each term gives:

\[ = 5x^3 + 20x^2 + 20x - 3x^2 - 12x - 12 \]

### Step 4: Combine like terms
Now, we will combine the like terms:

1. For \(x^3\): \(5x^3\)
2. For \(x^2\): \(20x^2 - 3x^2 = 17x^2\)
3. For \(x\): \(20x - 12x = 8x\)
4. Constant term: \(-12\)

So, the complete expanded expression is:

\[ 5x^3 + 17x^2 + 8x - 12 \]

### Step 5: Identify the coefficients
From the expanded expression, we can identify the coefficients:

- The coefficient of \(x^2\) is **17**.
- The coefficient of \(x\) is **8**.

### Final Answer:
- Coefficient of \(x^2\): 17
- Coefficient of \(x\): 8

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