To factor the expression \( 3x^4 + 7x^3 + 4x^2 \), we can start by factoring out the greatest common factor (GCF) of the terms.
The GCF of the coefficients \( 3, 7, \) and \( 4 \) is \( 1 \), and the smallest power of \( x \) present in all terms is \( x^2 \). Thus, we can factor out \( x^2 \):
\[ 3x^4 + 7x^3 + 4x^2 = x^2(3x^2 + 7x + 4) \]
Next, we need to factor the quadratic expression \( 3x^2 + 7x + 4 \). To do this, we can look for two numbers that multiply to \( 3 \times 4 = 12 \) and add to \( 7 \). The numbers \( 3 \) and \( 4 \) meet this requirement.
Using them, we can rewrite the quadratic:
\[ 3x^2 + 3x + 4x + 4 \]
Now, we can group the terms:
\[ (3x^2 + 3x) + (4x + 4) \]
Factoring by grouping, we get:
\[ 3x(x + 1) + 4(x + 1) = (3x + 4)(x + 1) \]
Putting it all together, we find:
\[ 3x^4 + 7x^3 + 4x^2 = x^2(3x + 4)(x + 1) \]
Thus, the linear and quadratic factors of the expression \( 3x^4 + 7x^3 + 4x^2 \) are:
- Linear factors: \( (3x + 4) \) and \( (x + 1) \)
- Quadratic factor: \( x^2 \) (since it's a degree 2 factor, not linear but still an important factor of the original expression).
Final factorization:
\[ x^2(3x + 4)(x + 1) \]