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

Answer

To find the constant term in the expansion of \((5x - 3)(x + 2)^2\), we can follow these steps:

### Step 1: Expand \((x + 2)^2\)
We know that \((a + b)^2 = a^2 + 2ab + b^2\). Here, let \(a = x\) and \(b = 2\).

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

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

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

### Step 3: Distribute \((5x - 3)\) across \((x^2 + 4x + 4)\)
We will distribute each term in \((5x - 3)\):

1. Multiply \(5x\) by each term in \((x^2 + 4x + 4)\):
\[ 5x \cdot x^2 = 5x^3 \]
\[ 5x \cdot 4x = 20x^2 \]
\[ 5x \cdot 4 = 20x \]

2. Multiply \(-3\) by each term in \((x^2 + 4x + 4)\):
\[ -3 \cdot x^2 = -3x^2 \]
\[ -3 \cdot 4x = -12x \]
\[ -3 \cdot 4 = -12 \]

### Step 4: Combine all the terms
Now we combine all the terms we obtained:

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

Combine like terms:

\[ 5x^3 + (20x^2 - 3x^2) + (20x - 12x) - 12 \]
\[ = 5x^3 + 17x^2 + 8x - 12 \]

### Step 5: Identify the constant term
The constant term in the expression \(5x^3 + 17x^2 + 8x - 12\) is \(-12\).

Thus, the constant term is:

\[ \text{Constant term} = -12 \]

---